…You could always try founding a colony in the middile of Greenland, too. Bury it in the ice, use greenhouses with sunlamps or hydroponics for food, dwell in a labyrinthine network of ice-tunnels, etc.
Your idea’s cooler, though. Especially if you don’t want to devolve into a subspecies of ice trolls.
I’m not even sure if ANYONE’S done mining from the seafloor, let alone a valulable element like Uranium.
Plus…even if it was technologically possible to get the Uranium, you might have problems with ‘certain people’ trying to…‘take’ it from you.
So, you might need to form a Marine detachment to guard the ships with reactors. One of the more reputable Mercenary firms would probably be able to set you up.
Let’s say we manage to mine it. I’ll write a revolutionary new mining submarine. How do we make it useful for reactor fuel? Is it realistically possible?
I’d highly recommend you check out a copy of the short story “Shark Ship” by C.M. Kornbluth. He described a sea-going society much like you’re looking for. His society was based on a fleet of sailing ships.
Some thoughts…
That Freedom Ship is essentially a floating condo on a permanent cruise. It doesn’t even try to be economically self-sufficient; it is assumed that residents buy their arartments, and than have enough yearly income from outside to pay for services and maintenance. That bein said, though, I think there’s a hotel/resort component as well.
As for Priceguy’s project…
You’ll be recycling bigtime: no chucking stuff over the side for you! You’ll need all those organics at some other time. Of course, once things are going, and you’re away from land, you’ll have a much smaller variety of things (packaging, say) to contend with in the waste stream than the typical city-dweller on land does.
Permaculture may prove useful. You’ll have to import topsoil or bedding media to start though. The permaculture I’m familiar with is land based; I don’t know whether you can permacultivate hydroponically. More? Google Bill Mollison and “permaculture”.
Doesn’t hydroponics require some fairly sophisticated techniques to create the nutrients?
There is much work being done with marshes as tertiary waste treatment facilities. This is connected with the design of living machines: series of connected water-filled vats, where each vat is a separate ecosystem, and together they purify wastes and grow crops. More? Google John Todd and Ocean Arks International.
You’ll have to keep your organic waste stream clean though: avoid getting heavy metals into the poop-to-fertiliser vats, for example.
There is much more of this type of information out there; it’s been a staple of “green” designers for around thirty or forty years. The Whole Earth Review magazine often has information about it; if you want more, I can supply many more references. 
Oh, yeah, you might be interested in this thread which deals with a big ass battleship in someone else’s novel. Here’s a couple of sites on a floating military base design that the government’s currently researching.
And Priceguy, I don’t think that the hydroponic chemicals have to be all that sophisticated to make. IIRC, there’s recipes out there for making them organically at home. Here’s a site on hydroponics. Another magazine you should probably look at, since it deals a lot with self suffientcy is Mother Earth News.
What is your background scenario?
Post-apocalyptic survivors?
Have them cannibalize rotting seaports for supplies.
*Colonists on an Earth-sized planet covered entirely with ocean? *
Then you can jump-start their energy needs with fusion reactors and hyper-efficient solar collectors. They should have arrived with some sort of ocean-bottom mining apparatus. (There are ways to extract minerals from sea water, but the energy costs are horrendous.)
Breakaway society continuing to live on the current Earth?
What is wrong with trade? If they want to be wholly independent, not relying on anyone and the technology level is our current one, they are screwed without a significantly larger population. (They will also face challenges from current governments regarding their ability to go sea-bottom mining without someone deciding that they have no right to do so.)(They are also going to need a defense force, given that they will need to load up on enough technology to make them very inviting to pirates.)
A separate ocean planet where society evolved to that point?
Sorry, could not happen.
(In support of your scenario, the naysayers who are claiming that seaweed and fish would be too “boring” are conveniently ignoring that most of human society has survived on fewer than a dozen plants and animals throughout most of history. Two or three varieties of seaweed and a dozen of fish would go a long way toward outstripping the diets of many, many cultures.)
Yeah, Tom, but what about those of us who’ve grown up eating large varieties of foods? It’s going to be difficult for us to adjust to such a bland diet permanently. (Admittedly, I could do it if it meant I was going to be living on Mars, but floating around in the ocean, knowing that there are cows out there, somewhere? I’m going to be wanting a steak, sooner or later, dammit!)
Would fusion power be of any use to you? Now only where would you get hydrogen in a big ocean and what would you do with all that helium you would produce, like how many birthday parties could you have, nah forget it, it was bad idea.
I think a fleet of specialized vessels would be needed, and maybe opperated like independant states but they have a common interest in trading. Maybe you could include rouge ships that will trade (or raid) with the landlubbers for vital materials.
A few off shore platforms or even underwater mining colonies might be good to throw into the mix.
You also need Lithium for Fusion.
Your source?
No. Mine, yes. But refining into useful U-235, U-238, etc. is very unlikely.
In the real world, this was done at places like Oak Ridge, TN and Hannaford, WA, etc. They all take up many square miles of land area, and generally seem to have left that landscape poisoned for the next few hundred or thousand years.
I don’t think you would have this much space available on your ship, and I really doubt that you would want such a dangerous area in close proximity to the rest of your living & food-growing area.
There may be more safe methods of refining uranium currently, but I still think they would take more space & more energy than your ship could spare.
But don’t nucleur fuel rods last for decades? With a few spares, you could go 50-60 years. Heck, most countries in the world haven’t been around that long.
Not with my fusion reactor, which model do you have?
My post got eaten, but not by the hamsters. That said, the short version:
Boy surviving in life raft (and good story): The Life of Pi, by Yann Martel.
People surviving in general: Kim Stanley Robinson’s “Mars” trilogy (of four). Includes a group who break away and carve out a biosphere under a polar icecap.
My contribution: bottom-crawling robots for mining and drilling of oil. Visit periodically to collect the harvest. Possible plot point is discovery of robot by land-farers.
Thought: multiple boats good until a storm tears them away from one another.
But I wrote it all much more eloquently the first time.
Well, if you mine the Uranium yourself, you might just bring it ashore to be refined. Countries like South Africa, India, or Pakistan probably have the means to do it, and might refine it for you for a large enough payment and/or a “cut” of the processed material.
Just don’t give ALL of the Uranium to them, all at once.
Oh, and there’s a “slight” possibility that you might get “visited” by the SAS, SEALS, GIGN, Spetsnaz, Shayetet 13, etc., as a lot of governments wouldn’t really like having everyday folks running around with refined fissile material (Spoilsports).
Didn’t that happen to the guy who founded “Sealand” or whatever the country’s called that’s an old gun platform?
Manganese is mined routinely from the seafloor… also, uh… what exactly did you think oil rigs were there for?
Sealand is visited on a regular basis by Royal Navy vessels and a few seamen disembark to make sure nothing fishy and Bond-villainesque is going on, IIRC.
We’ve got quite a clever little ocean-civ going on here… Priceguy, you may want to consider writing in the concept of all the inhabitatants being Dopers 
If the crop being harvested doesn’t go through a “middleman” (cow, pig, chicken, etc.) before being consumed, then the average human needs 1/4 acre a year, AFAIR. If you feel the need to feed your crew beef, pork, or chicken in addition to the fish and seabird the requirement jumps to 3 acres. With recent genetic engineering and improved strains of grain the yield might be improved. If this is to be a science fiction story then I think it would be reasonable to have a moderate crew, especially with several ships, provided significant advances in crop technology have happened, and if the only meat the crew eats doesn’t come from the boat. Also, the numbers above are for a “normal” growing season. If your ships are in a tropical zone you should be able to grow food year round. I would say it would be reasonable to feed 10 people for 1 acre, if it takes place in the future far enough. I don’t know how that would translate to an aircraft carrier or a barge, but with enough ships I’ll bet you could do it.
Fusion would probably be your best bet for your energy needs, especially if it takes place in 50+ years. Even if in reality you do need certain unobtainable ingrediants (lithium) to make it happen, you might get away with some literary license and just say they extract the material from sea water. Consider your audience. Are they dopers or just your average dope? Who questioned Crichton’s dinosaurs? He was missing a very vital ingredient: getting the DNA into an egg (which apparantly has since been solved with modern day cloning) but most readers didn’t even realize that it was missing.
My grandfather did subsistance farming on 49 acres in north mississippii all his life, the acreage provided most of the support for his family (self, wife, three children -actually 4 but one died at birth).
But - he raised about 10-15 acres of cotton which provided the cash for shoes, sugar, coffee etc. Total self sufficiency is damn difficult on land.
As far as keeping them in a good mood, as you referenced in your post, I’d recommend marihuana, + it is excellent cash crop (much better than cotton)
Seconded, in the ocean there are dissolved particles of almost every material, and while it is very expensive and not cost effective to do so, its been done in the past.
I think it was done at a texas plant during wwII (?) to extract magnesium (?)
Either way, IMHO you could stretch this technology to (at present) unattainable levels, without raising many peoples brows. Especially if you used true real world examples.
Two thoughts:
Seagoing pirates are still a problem in some bodies of water, which many people find surprising in this day and age. (CNN story from less than two years ago.) So depending on where your boat will be sailing, you may need to consider a rather serious self-defense capability. And how does this impact the ship’s social structure? Is there a dedicated armed services, or is it more like Switzerland where everybody is considered a potential soldier and needs to learn how to fire the railgun? Also, where and how does the ship procure its armaments?
Also: Just how self-contained a colony are you talking, and over what period of time? I’m specifically thinking about reproduction and population replacement, and I’m wondering if 300 people is enough to provide a reasonable basis of genetic variation. Especially considering how people tend to segregate themselves, you could be looking at an inbreeding problem over several generations. Perhaps the ship’s planners took this risk into account, and recruited its passengers from all over the world in order to maximize the population’s DNA library. Somebody with more knowledge of human genetics might be able to offer more information about how many people would be required to provide a viable breeding group and, if 300 people is right on the borderline, whether a program to test people’s DNA and match mates with the greatest genetic variation might be warranted.