If you want to keep the fleshy bits of your space elves, steer well clear of pretty much any kind of nuclear power supply, fission, fusion or annihilation. Highly penetrating x-rays and gamma rays you may be able to shield against (though I wouldn't bet on it) but the spray of fast neutrons you can expect to find flying out of any fission reaction and many fusion reactions (yes, even proton-boron fusion, where about 0.1% of the reactions will shoot out nice, fast, highly-penetrating and highly destructive neutrons) will be basically unstoppable and everyone will die of cancer before arriving at the new world. Even if you have super space medicine to fix cancer, those brains are gonna be pretty fried a lot of cells are gonna die. Not good news.
But to flip it around, how much power do you think you really need? "nuclear reactor" sounds like the sort of thing you'd want to power a rocket engine, or an energy weapon, not a person. A human body has an average power use of about 100W. An efficient robotic chassis with similar performance could be driven with modern batteries and recharged from time to time (daily, probably). On a spacecraft designed for such things, you might have charging points pretty much anywhere and everywhere people might congregate or rest. Super future batteries or fuel cells could either be recharged much less frequently, or provide much more power, if you can think of something to do with it all.
Remember also that a lot of the power draw will be moving around, but on a non-relativistic spacecraft (or even a plausible relativistic one, to be honest) there will be little to no acceleration from thrust so the only gravity forces you have to work against are artificial ones that you can dial up or down to your heart's content. Just turn the spin decks down and relax in microgravity.
(also if you really, really want fission power, be aware that fancy isotopes with small critical masses tend to have half-lives much, much shorter than your projected thousand year flight time. bring a big breeder reactor and a good supply of parent isotopes to work from)