I think you should be able to capture, feed, and breed rabbits for food and fur.
You should be able to construct a burrow that holds 6, and place a male and female inside and them breed every 12 real life hours. Once they breed they produce two baby bunnies which grow into full adults in a full real life day.
The burrow should have a green bar similar to the fuel/water gauge like on the side of the ore smelter. You should be able to put in mulberries, apples, and wheat and it fill up. Should the feeding bar ever run out there should be a real life 12 hour countdown until they starve to death.
Or, another option would be to make a "pellet compresser" machine that takes apples, wheat, and mulberries which converts them into pellets that you can fill the feeding bar of the burrow with.