Nomadic humans travelled in a single generation. These flies need to be DTF in order to finish their journey.

The Biosphere 2 project was an attempt to see what the smallest self-sustaining ecosystem (that would feed humans and recycle air/water) would be. For a crew of 8 people, the area of plants, crops & wetland covered 3.14 acres.

https://en.wikipedia.org/wiki/Biosphere_2

If I recall correctly, it ran into a number of problems, and they abandoned it when oxygen levels dropped too much. So I don't think this counts as having mastered it.

They were also quite naturalistic though too, they could have packed more into a smaller area with vertical aeroponics etc to decrease the area.

Not necessarily what you really need is enough excess mass of critical elemental components to make up for any gaps in the recycling loop(s) between stars where you can resupply from asteroids.

Made me start wondering if supplies could be picked up in route. The oort cloud (https://en.wikipedia.org/wiki/Oort_cloud) extends most of the way to the next star and presumably extends a similar distance away from the next star. Missions would need to be sent out in advance of the ship to start collecting and making fuel. It could then be accelerated up to catch the generation ship. It seams easily plausible in a science fiction sort of way.

Generation ships can't slow down. They don't have the fuel to slow down and start back up, only to stop at the end.

Also, the supplies should be available in the home system. It costs just as much to send out probes and accelerate the supplies as it does to send the supplies with the ship.

The only exception is that don't have water as fuel, and could travel slowly to the closest icy object, and then do the full burn to speed. It would add years though.