What do you think will be the most significant problem going to even the nearest star?
In my opinion, based on reading a lot of articles about the issue, my prognostication is this: in the coming century or two, I think we will be surprised by the power of future space drives, mainly in the area of anti-matter drives. Here is my reasoning: while the idea of humans manufacturing anti-matter is within the realm of the possible, it would be extraordinarily expensive for us to make enough for an interstellar flight, I am thinking maybe 1000 Kg or so, maybe less. But it has already been discovered there is a huge supply of anti-matter already present in the solar system, due to the ratio of matter to anti matter in the universe in general, about 10 billion atoms of matter to one of anti. So it has been shown a collector, a mesh charged properly and an antimatter trap in the center, maybe a klick long, like a giant soccer ball (football for you Euro's) but no outer skin, just wire where the pattern is, will collect kilograms of the stuff if over the course of a year or two.
The interesting thing about all this is the engineering on antimatter rockets have pretty much already been done, they are much further along than it's little brother, fusion rockets, and a thousand times more powerfull potentially. If you give a significant amount of antimatter to a present day rocket scientist, he or she will be able to built a nice rocket right now. That said, if there was a large amount available, then it would have the capability to accelerate at about one g or so continuosly for years on end, then halfway to AC, decelerate the rest of the way. I think it will be shown we can do exactly that, maybe in 200 years, maybe less.
But it looks like the biggest problem will not be food supplies, hydroponics, or any of that. It looks like maybe the show stopper for human flight anyway, is the sheer radiation of just traveling at even half the speed of light. There was a design for an interstellar craft designed in the '60's that fed a bunch of small atomic bombs to a giant pusher plate with shock absorbers that would give a huge pulse of accel, say once a minute or so, don't know the exact timing, but it was calculated to be able to get up to maybe ten percent of C. Now you would think the radiation shielding would be a big problem to keep it away from the humans and it would be, but that could be designed in alright, we now know stuff like ordinary plastic is a very good radiation shield and such. The BIG problem is not the radiation coming from the back by the engine, whatever it may turn out to be, but the radiation coming from in FRONT as you get up towards relativistic velocity. Remember, as you increase in velocity, the wavelength of incoming radiation is blue shifted in front. So infrared may become Xrays when you get close to C. That radiaton may be the killer of interstellar flight, way to powerful to be deflected by magnetic fields or plastic shielding, at least thats what it looks like now. But if they get around that problem I have no doubt we will be an interstellar going species in a couple hundred years IF we live through the coming climate shifts that may kill high technology in the first place. That is a much bigger challange right here and now than getting enough anti matter for a flight to Alpha Centauri, surviving the coming climate battle with our technology for spaceflight still intact.