[api]

Food can be recycled pretty effectively, and if were that good at biotech I assume we could improve on the current state of the art.

They already recycle water very effectively on the ISS. It's the machine that "turns yesterday's coffee into today's coffee."

Of course if we were that good at biotech we could probably hibernate a good chunk of the flight time too. Might be necessary to wake up periodically to reset the body, but you could probably hibernate most of the duration. Maybe you'd do it with some kind of weird circadian cycle with extremely elongated sleep periods, sleeping like 10X-100X as long as you are awake. During each wake period you check to make sure everything is working properly.

You would not need a Bussard ramjet for the long duration flights I'm thinking of. A nuclear thermal rocket could get you a good deal past solar system escape velocity. Nuclear pulse propulsion could get you up to at least single digit percentages of the speed of light if you didn't mind a little boom-boom. Then you just cruse along on an interstellar transfer orbit until you do a retro-burn to enter the destination star system a few tens of thousands of years later. These are all technologies that are already feasible at least on paper. No new physics is needed.