I am not sure I see the reasoning here. Without a limiting time frame I don't see why we shouldn't solve most of the problems:
-radiation. Solvable by bring enough materials with you which currently is limited by launch cost and launch cost only. The energy requirement for getting stuff into orbit isn't really all that much so the problem is purely technological and not a problem of physics. With half a century of space travel so far there has to be limits there still possible to push.
-lack of gravity. Large enough acceleration and it is solved, but there we might bump into what's physical possible to achieve. But, tether two crafts together and send them into a spin and you are set. Just a matter of big enough spacecraft, see above. I suspect a 10 km rope would do wonders.
-supplies. As of now, humans still need an ecosystem to function. We are less and less dependent, but we are not there yet. However, there is no reason we can not achieve an artificial ecosystem where food is either grown on the waste or synthesised directly. Carbon dioxide capture and oxygen production will regardless be a side effect.
-energy. Some sort of nuclear reactor (be it fission or fusion) has to be brought along and fuel to feed it. But we should be able to do that today, if we really wanted to, so no problem there either.
The problems that might be unsolvable is time or speed. Even at light speed, you are correct, we will not reach very far in a life time, just a second, some guy is calling about relativity. Where was I. Yes, if we achieve on the outside, half the speed of light, we will not get very far in a life time. So either we have to extend the life time or we have to send our children even further and so on.
Becoming intergalactic might also be out of limits. The distances can become so large that without physics yet to be known can come to our aid, we can for all practical solutions be stuck in the Milky Way.