I had kind of wondered about this actually.
I'm not sure the genetic diversity argument holds much weight though. There are very small islands with no real outside contact that have maintained populations for a very long time. There is little variation in the physical features of the inhabitants. Apparently they make for interesting genetic studies. Also, how do we exist? Whether you believe in a creator, or you believe in evolution, either theory seems to suggest life as we know it came from a rather small genetic pool. It seems very unlikely that life would be spontaneously created in many different interbreeding forms all at the same time.
I imagine 200 people could create a self sustaining colony. I'm not sure what population size is typical for isolated and independent islands, though if you read up on Pitcairn Island, it currently has about 50 people among 4 families. It seems to have been founded by less than 30 people, and had a max population of over 200, but most have emmigrated elsewhere.
For comparison, I hear aircraft carriers have a population of about 5000.
So yeah, I don't think a gene pool is strictly necessary for 200 people. It probably just seemed like a good idea.
As for supplies, a spaceship must travel much further and for much longer than a sea vessel, with no chance of getting resupplied. At best, they could harvest stuff along the way. However, if they're travelling through deep space, there probably won't be too much to harvest. Plus, getting all the supplies into orbit if a much more difficult and expensive task. I would imagine crew sizes would be much smaller. I have no idea how big the physical dimensions of the Outpost starships is, but a population of 200 seems pretty massive compared to typical space going vessels today.
I would guess such a spaceship might very well be smaller than a battleship. I would also guess a lot of the space was taken up by fuel to make such a long voyage. I doubt computers would take up much space. I doubt sleeping quarters would be that much of an issue. Most of the crew would be asleep the whole journey, and they'd probably be packed in fairly tight. Not many people were awake on the ship, or for very long, so you'd need minimal space for working/eating and the like. The majority of the population space would likely be taken up in stasis chambers. It may be a reasonably significant part of the ship, but it's not going to be all that big. Also consider that the space taken up by an individual doesn't change if they are working or sleeping. That space needs to be somewhere, and if they are sleeping, you don't need additional space for the other things people do.
One of the points of hiberation is to reduce food and oxygen demand. I suppose if the systems really could manage suspended animation in the most thorough sense, food and oxygen demand should drop to zero. Particularly if aging is to be halted. Things like oxygen and nutrients should be in minimal demand for the duration of the flight, but a good store probably needs to be kept to support people when they land and are woken up. Afterall, you don't know what kind of environment you'll find at the end up such a long trip into the unknown.