Perhaps your story could begin in Kansas, on a Saturday, and a huge tornado lifts an entire Home Depot and shoots it through an opening in the fabric of time. Then you would have a bunch of tools and supplies, a large gene pool and most importantly, a fat advance on the rights to the story from Home Depot.
Thanks, I know that, but what I don’t known is how many.
I know … that’s what makes it challenging. However, many people carry Swiss Army knives at all times. I carry a complete automotive tool kit in my car. Tom Hanks in Castaway did a lot of cool stuff with an ice skate.
I recently read 1632 and 1633 by Eric Flint, et al, as well as the Nantucket series by S.M. Stirling. (I enjoyed them very much, by the way. Two thumbs up.) But in a certain sense, I thought they cheated by moving an entire modern American city into the past. So what’s the minimum number of modern Americans who could get shot into the past and survive?
Yes, all of these skills and many, many more. Rome will not be built in a day, nor will it be built by slack-jawed couch potatoes … unless those couch potatoes watched a lot of the History Channel. The question, though, is not what skills are required, but rather, what modern American professions possess enough of those skills to survive the experience? For instance, Navy Seals are given thorough wilderness survival training, so that’s one. A Renaissance Fair blacksmith might know how to prospect for coal, iron ore, and limestone, and combine them to make steel tools. And so forth.
The trick, here, is to figure out what professions these people have, and then figure out who they are (characterization), and then figure out why they would all be in the same place to get shot into the past.
On their way to work a botanist, architect or metalurgist might have some darn useful stuff on their laptop.
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Of course, batteries would be an issue. 
I’d still like an architect along though and skilled low-tech tradesmen.
Don’t bet on it. Even if you extracted a professional blacksmith from the actual Renaissance (which would be an interesting idea), sure he’d know how to compose his metals out of those ingredients, but probably not how to actually dig them up. Things were becoming highly specialized even then - miners harvested ores, blacksmiths melted them and hammered them into stuff. Two separate disciplines, each of which was a huge skill set in it’s own right. And while there are working blacksmiths in the modern era, and some of them might be interested enough in the history of their profession to work at a Ren Faire and do it the old-fashioned way, I feel pretty confident in stating that it would be highly, highly unlikely that any of them would have a clue about how to prospect for anything.
I guess I should reiterate, also, that the answers should be different depending on whether these people are dropping into a virgin wilderness or a wilderness populated by other humans.
Thanks again.
I think you’d need around 25 people, from many different professional backgrounds, a high percentage of whom also have some statistically unlikely but coincidentally useful hobbies, in order for them not to die from hunger or exposure pretty quickly. Of course it depends on where on the continent you drop them. I have no idea if 25 people would constitute a varied enough breeding group, but I will assume so unless someone who knows about genetics offers a more educated opinion.
Because the skill sets for survival at different levels of civilization vary so widely, the main body of the group would be dependant on a relatively small percentage of the group and their specialized knowledge. The main body of the group would have more specialized skills and would contribute more at later stages, after things stabilize. I mean, who cares about paper until you’ve got the food supply well-established? And if one person has a notebook with them, then you’ve got enough paper to last quite awhile if used judiciously. But judicious use, in the eyes of the survival-minded group, would probably be for starting a fire, not documenting their adventures. For example, even if someone knows how to build an usable bow, they could starve to death unless they were also good enough with it to hit something as unpredictable as a rabbit. So, you need people who know how to build traps and/or can build bows and shoot accurately with them, (a statistically unlikely but not TOTALLY improbable hobby). And enough of them to successfully provide sustenance for the rest of the group. I forsee them eating a LOT of stew. 
Clean water might not be a problem. There weren’t a lot of pollutants back then. It would be EXTREMELY helpful (and not too implausible) for someone to have matches. Then you’d only have to start one fire then keep some live coals going. I’ve read that this is most likely how early man got fire to begin with - found a log that was burning from being struck by lightning, and just kept the fire going. There are even ways to transport live coals without them burning through the container. If they have fire, they can boil the water, if they have something to boil it in. I’d say the more metal they can bring with them, the better off they’ll be. It’s incredibly useful and can often be reshaped without specialized tools.
You need someone who knows what plants of the time and place are edible, which ones can be used for herbal remedies, and maybe how to make a poultice. You definitely want someone with a background in emergency medicine, who can set a bone and stop someone from bleeding to death, treat stuff like gangrene and frostbite, or do an amputation if necessary. Shelter might not be that difficult if you drop them somewhere with a temperate climate, during the right time of the year. Someone who was a very enthusiastic Boy Scout, an obsessive camper, or has survival training would be a downright necessity.
Later on you’d want a geologist or metallurgist to help find those ores. A geologist might also be able to tell you which soil would be the best for planting, though a farmer or botanist might be able to do the same. You’d also want individuals that know about things like irrigation, tanning and curing hides, food preservation (smoking and drying meats, pickling vegetables), woodworking, blacksmithing, spinning and yarn-making, weaving cloth, glassblowing, ceramics, etc. Then once they’ve got all that worked out, somebody might want to try making paper. 
Michael Crichton’s Timeline and Diana Gabaldon’s Outlander series also deal with the problems of people who find themselves outside their accustomed time. They are fascinating stories, filled with problems the average person would never consider. I wish you good luck with yours! Maybe you’ll let us read it when it’s done?
“Shop Smart. Shop S-mart. Got that!?”
After reading that other thread, I’ve changed my mind about things. If you give these people a stock of seed for modern cereal grains suitable to the area in, they have a chance of retaining some technology, given the right skillset. If you don’t, all their time is spent procuring food, and within two generations you’re right back to stone age hunter gatherer society, cultural transmission of knowledge will cease, and the few that survive will be essentially indistinguishable from the other humans who actually existed at the time.
For pure literary entertainment, I think I’d go with a full-blown SCA meet getting teleported. A sort of ``Yeah, you guys enjoy this stuff when it’s make believe, but can you actually hack it for real’’ kind of deal… It wouldn’t be too big a stretch to have that group be nice and diverse in terms of other hobbies and professions. I also think for the purpose of your story, the hobbies are going to be more useful than the prosfessions. Professions are so specialized, I don’t think any of them would be fully useful.
Now that’s interesting …
Let’s extend the OP to include hobbies as well as professions.
Look, if you dump people with only the contents of their pockets into the wilderness, they are going to be resorting to cannibalism in only a few months.
Even to sustain something as complex as a hunter-gatherer society requires a huge knowledge base. Your average hunter-gatherer is a knowledge professional every bit as skilled as the most advanced lawyer, computer worker, or doctor. The only difference is that everyone in a hunter-gatherer society learns pretty much the same types of skills, with perhaps some specialization based on personal tastes.
Remember the last time you walked in the woods? How many animals did you see? Without a rifle, how many of them could you have caught in a day? How many plants that were edible did you see? You may have been able to gorge yourself on berries for a few weeks in the summer, but what about in spring, fall or winter? What kind of shelter could you make, without tools? Even if you manage to survive for a few months eating bugs and leaves, what are you going to do when your shoes wear out?
What if you cut your foot? What if it got infected? What if you ate something that didn’t agree with you? Even if you might get better soon, how are you going to feed yourself while you are laid up? And if you can’t feed yourself, how are you going to be strong enough to heal?
The idea that a bunch of hobbyists with the right combinations of skills could get together is kind of silly. In a few months your amatuer blacksmith is going to be hungry enough to start cracking open his neighbor’s skulls to feast on their calorie-rich brains. He’s not going to be searching for coal seams and iron ore, he’s going to be starving to death. Hunter gatherer societies can’t support specialists like blacksmiths without a huge surplus. Within a few generations your band will have all the technology of Tasmanian aborigines, if they survive at all.
I think it takes a group of at least 50 people or more to prevent inbreeding, probably more in the long term. I do know that early native american tribes would regularly bring in fresh blood from other tribes through intermarrying. In any case if these people are on their way to work and would presumably be in their cars do the cars get transported as well? Even if you can’t drive it it is a fairly decent shelter and source of materials. A tankfull of gas means you have and easier time starting those first few fires and a limited supply of explosives. I think there was an episode of sliders where the world was going to be hit by a solar flare or something and they had to use the portal to get a small group to a dimension where there were no humans and they faced exactly the question you are posing when choosing who would go.
One thing that has been overlooked is the specific date and location- 14,000 bp, North America. On this date, glaciers cover the top half of the continent, so our settlers will have to be fortuitous enough to have been displaced somewhere south if they hope to engage in any significant agriculture or mining. At the same time, they’d be contending with a menagerie that includes mammoths, mastodonts, glyptodonts (giant mace-tailed armadilloes), sabretooths, and elephant-sized ground sloths. There would be horses but none domesticated, of course. The Pleistocene Overkill Hypotheses suggests that when humans first arrived on the continent, this abundance of large game showed no fear of the newcomers, who partook excessively of this great bounty and drove them to extinction. So in that sense, our party may not worry about a steady supply of fresh meat.
If it’s an SCA group, the odds are tilted quite a bit in their favor. In addition to the higher than usual tech-geek quoteint, I’ve run into a surprising number of folks in the SCA who are avid outdoorsmen.
I think location is more important than anything else. If you’re dropped somewhere with a food supply, or the potential for same, it’s all good. If not, things start to get dicey.
As for “tool in your car”, I have at least 2 knives in my car pretty much at all times. Plus my tool kit, which is not great, but would give me screwdrivers to use as spear-tips if need be.
I think it would be a matter of time, and society would revert to pre-industrial, but they would survive, no problem.
As for how many… well, depends. Humans are fickle.
But hell, I’d volunteer.