If it all takes place in another, huge, solar system, and they don’t have FTL, then how did they get there in the first place?
Well, they could have slow FTL, or near-light transportation. So Earth might be in the center of a 100 light-year colonization zone where it takes weeks, months or even years to move between some of the star systems, but once you’re in a system you can zip from moon to moon in anywhere from hours to days.
I’m not saying this is the way it is, but in a lot of ways it makes sense. In ‘out of gas’, they take two shuttles and go far from the ship in order to widen the ‘search area’. Now, unless those shuttles are FTL, anywhere they could go in a matter of a few hours would be so trivially insignificant against the scale of interstellar space that they wouldn’t even bother. Plus, ships just ‘happen across’ one another - it’s happened in more than one episode. That would be an argument FOR a crowded universe. Take ‘out of gas’ - they were obviously in transition to a distant destination. If it’s another star system, what are the odds of coming across another ship out there? But if it’s on a direct orbital line from one planet or moon to another, suddenly it seems plausible.
Um… yes, it’d be easy for them to plot a course outside of normal flight paths in “Out of Gas.” Space is three-dimensional, although (at least in our star system) the planets orbit across more or less a single plane. They could fly up out of the plane of the ecliptic and over. It would use more gas and they’d be hell and gone from a rescue, and it’d take a lot longer, but sure, it’s possible.
Or, if you prefer a alternate explanation, there are no particular flight lanes because the planets are all in motion. In our system, for example, a year on Saturn is about thirty times as long as a year on Earth; over the course of a year of flying back and forth from Saturn to Earth you’d have to chase Earth around its orbital path. For all practical purposes over that 1 earth-year, Saturn is virtually stationary. (It would more 12 degrees in its orbital path as Earth moves 360.) There would be no continuous “space lane” as such, although there would be an ideal Shortest Path.
Plotting the less-traveled course would be possible at certain times of the year depending on the positions of the other planets.
Mars is a more realistic comparison: its year is not quite twice that of Earth’s. Flying back and forth between the two is an endless game of dodging around the sun. Either that, or you would see an increase in space traffic when the two planets are in the same quadrant of the solar system.
My further one-star-system suspicions are supported in the pilot, where Mal is able to tick off on one hand the places where their goods could be sold. Surely if they were able to reach other stars they would have far, far more markets for their merchandise, some of them not even under the thumb of the Alliance.
That’s the biggest point for me, though. If there were a place where Mal could fly away and escape the Alliance, would he hang around here and deal with people he doesn’t like (Badger), people he doesn’t trust (Patience), and people out to kill him (Niska)?
The cowboys-in-space metaphor is a good one. Persephone would be… oh… Kansas City. Or maybe St. Louis. Whitefall would be like Dodge City.
Could every moon in a solar system be inhabited? Not with the technology we know. Even if we could solve the problem of creating Earth-like atmo, temperature and gravity, we’d have to find a way to shield Saturn’s moons from Jupiter’s huge asymmetrical magnetosphere, right? Once a year Saturn’s moons would cross through Jupiter’s influence and that’d be tough on any colonists living there (not to mention any living on Jove’s own moons!).
Or those were simply the places he knew that would be interested in their goods, ones that aren’t under the thumb of the Alliance. Serenity spends most of her time away from civilized worlds, I’d take “civilized” to mean “Alliance-controlled”.
If it’s not one big star system, then why have so many of the places they’ve visited been moons? Why would the writers bother confusing the average viewer like that?
Also, if they have space drives so good that they can move from star system to star system in a matter of hours or days (even better than Star Trek!), then it does seem strange that all these good and bad guys would be so crowded together. With ships that fast, humanity could spread out through the entire galaxy, and never see one another.
But maybe the most obvious explanation is that Joss isn’t enough of a science fiction fan to get the nuances of all this, and since the story is more about people than high tech, they’re just going to leave the details hidden and focus on the people.
I’d say that you guys are thinking too much about the travel times except I once developed a whole justification for the use of horses:
See, the way people ended up populating the outer planets was the Alliance did terraform them to an extent but didn’t want to pay too much and left them as somewhat less than garden spots, looking strikingly like Southern California ;). Immigrants were dumped on these planets with little more than what they could carry and were told to sink or swim. (This was all explained in some episode or another.)
Anyway, you are poor and moving to farm a homestead far, far away. You can barely afford to pay transport for you and your family, much less a tractor, but there wouldn’t be anyplace you could get parts for a tractor, anyway. Cars are totally out of the question; life will be very low-tech, possibly for a few generations. Your equipment (plow, wagon, etc) will be built from locally available material, probably wood for its availability and easy working by any schlub with a saw, but you need a power source that doesn’t require special fuel or expensive parts and can be locally produced. The answer? Animals! They can feed and take care of themselves and they reproduce readily. If the colonists chipped in on transporting a few mares and some frozen embryos they’d have all the motive power they’d need in a few years.
Then there are the 4-wheel ATVs that make up the lion’s share of motorized land transport. An excellent idea because…(a click as drop turns off his brain)
Damn! I knew Joss was smart, but not that smart! 
Because the effort to terraform an entire planet is orders of magnitude more expensive, difficult and time-consuming than terraforming a moon the size of, say, Titan or Ganymede? Even after terraforming, most of those moons seemed to have plenty of empty spaces.
Dropzone: Yes, that’s the basic process they seem to be modeling–drop the colonists on a planet with a herd and some basic tools, and they have to make do with what they can. You’ll notice that most of the places that have higher-tech stuff have a fairly high-tech reason for it–it’s a space port, or it’s the train that carries workers, supplies and ore to/from a mine that they’re operating, even though it kills the people who work in it, so it’s gotta be pretty valuable… If you’re someplace where there’s a lot of ships coming in every day, it’s easier to get parts and power cells from the core planets where they’re making them. If you’re on one of the outer planets, no, not so many parts are available…
And it worked, too, Miller. You guys think you’re obsessive sussing out the details of travel in the Firefly universe? The gun fans are just as bad! http://forums.prospero.com/foxfirefly/messages?msg=1040.1
They’re not confusing the average viewer. They’re leaving the details vague in order to avoid confusing the average viewer, who doesn’t care about interstellar velocities and orbital mechanics and the physics of gravitation. But by doing so, they’re confusing us, the community of people whose Straight Dope membership overlaps with our interest in SF, which makes us smarter than the average bear. 
Serenity, part 1. When the crew and passengers are having dinner, and Mal is explaining to the one guy (the passenger who wasn’t Simon or Book) what life is like on the frontier worlds.
Because the ships don’t have unlimited range. They need fuel to run on, as established in the pilot, although I don’t think they ever say exactly what they use for fuel. They can only fly so far before they need to turn around and go back and refuel, and they can only do that at a colonized world. Sure, the frontier is always expanding, as new worlds are colonized and new spaceports are built on them, but colonizing a world is both tremendously expensive and time-consuming. It would take generations to go from discovering a viable new world to building up the population base and infrastructure enough so that it’s a viable colony and launching point for the exploration of the next closest star. Space is limitless, which means it takes a long time to fill it up. Humans in the Firefly universe haven’t been spacefaring that long, and they’ve only got seventy or so colonies going. Of those, only a certain percentage are going to be the sort of planets that need the services of a ship like Serenity: the border worlds, where the Alliance prescense is weak and the black market thrives. Since Serenity can only make a profit running in those circles, its not at all surprising when someone they’ve crossed before shows up again. Especially when they’re someone like Niska, who’s probably got a piece of the action on every backwater planet with a large criminal underground.
As for encountering ships between systems, part of that is dramatic liscense, but part of it is also that, unlike planets, solar systems tend to maintain the same distance and position relative to each other, at least when measured in a human timeframe. Since fuel is expensive, most ships will take the shortest distance between two systems most of the time: in effect, creating a “space lane.” Out of Gas has them taking a roundabout way of getting between two systems that was not widely used, unless you had a reason to want to avoid the Alliance. Mal was lucky in that he wasn’t the only ship captain who knew about the alternate route, and he ran into that ship that had the part he needed. (Of course, the reason they were taking the long route themselves was that they were pirates.) It’s still pretty implausible, but less so than the idea that a non-FTL ship could get so far away from a solar system that it wouldn’t be able to contact anyone in-system on radio, even if they were traveling at a right angle to the plane of the elliptic, and still have enough fuel to get back to an inhabited planet.
Oh, you’re no fun anymore. 
Hey, there’s the one nitpicky thing that kept bugging me throughout the episode. That money has got to be danged valuable. A little sackful of it buys enough fuel to fill the tanks on a starship! Again, like so many other things, Joss isn’t trying to make Firefly: the Role-Playing Game, so he hasn’t answered a lot of little things like that, but if a small sackful of coins buys a couple dozen cattle or a shipful of fuel, what the heck kind of coin do you use to buy a pencil? Or is the future like Sam’s Club, where you can only buy a dozen gross of pencils, and then they last you for years?
Maybe the bag was full of bars of latinum instead of slips.
What am I, Miller, chopped liver? 
Maybe there was a personal check in the little leather bag.
Wouldn’t a sackful of $100 bills buy enough diesel fuel to fill up a small cargo ship?
I was more impressed by those food ration bricks. Didn’t they say one brick feeds a family for a week? One quart of pure fat has only 9,000 Calories, and I don’t think there is any substance we can digest that has higher energy density. 9000 Calories shared by 3 people (2 isn’t a family, is it?) over a week is only 430 Calories per day. That hardly seems adequate, especially on a frontier planet.
A month if you don’t like the kids.
The communication thing doesn’t bother me at all. Unless you’ve got gigantic telescope arrays pointed right at a ship in interplanetary space, you aren’t going to hear any distress calls. Plus, if we posit a large star system, the planets could be many light-hours apart.
The one planetary system theory does explain some of the terminology - the ‘core’ or ‘inner’ worlds, and the ‘outer planets’, where everyone seems to live on moons. That describes our solar system (not saying it is ours, but it’s likely other solar systems are the same - inner rocky worlds with all the heavy metals and resources and wealth, and colonized moons around outer gas giants.
I think we’re maybe all so conditioned to science fiction involving flight among the stars that we’ve just accepted that that is what’s going on. But if you read ‘classic’ SF like Heinlein, he has a universe where there are terraformed Jovian moons that are frontiers (even with horses - much like Firefly), and rich inner worlds that have advanced tech.
Let’s try it this way - is there any hard evidence in Firefly that they are NOT in one giant star system? Any references to faster than light, or different suns, or anything like that?
I suspect that humanity chose this particular star system to colonize precisely because it did contain a high number of probable planetoids. Why bother taking a sub-light flight of, I don’t know, 250,000 years or so only to arrive at a dinky one-habitable-planet system like Sol’s? What if that planet turned out not to be suitable?
There may be other inhabited solar systems in the galaxy, but I suspect they’re like the one we’ve seen: lots of habitable worlds, or at least lots of worlds that could be made so. Two such solar systems near one another doesn’t seem very likely. The distance to another inhabited star system would probably be prohibitive.
Without FTL, how did they get to this star system? Easy, I’ll answer the question with a question: what did Simon take River out of in the pilot?