Do worker Aniimo stop working without food?

The food sits in a feed box, and that box is what you are actually managing. Your Aniimo draw from it while they work, and when it empties they stop. Nothing breaks and nothing is lost. Production simply halts until you put food back in.

This matters because of what the Homeland is for. It is a solo base with a farming system, and the point of assigning Aniimo to work there is passive output while you are off doing something else. A stalled feed box turns that into nothing. The whole time you were away, the base was idle. Feeding is also what holds work efficiency up, so the feed box is not a switch you flip once. It is the thing you top up before you log off or head out.

The same applies to Aniimo working in the RV. They have to be kept fed too. Upgrading the Homeland RV to level 2 unlocks worker automation, which is the upgrade to chase if you dislike the babysitting. Until then, the routine is simple. Check the feed box on your way out of the Homeland, every time.

Not settled: How long a full feed box lasts is not stated, so you cannot yet plan a long trip around it. Check before you leave rather than guessing.