I guess it would be possible. Would they be in air conditioning? Chickens have a hard time laying eggs when it is too hot. I know nothing about goats. They have tons of energy. I can't imagine them being cooped up in a small space.

Welcome on the journey to homesteading!

It is completely possible to do exactly as you suggested. You would need electricity in order to ensure your chickens received 12 hours of sunlight for egg production. You should probably lay back on the lights during the winter and fall to reflect sunlight hours in your region to give the chickens a break.

In selecting chickens, I would go for fuller bodied breeds. This would allow you to keep the hens as layers and slaughter the roosters for a decent meat yield.

Goats would be trickier. If you are set on not allowing the goat outside access I would focus on selecting a dwarf breed, or at least smaller breed. I would keep a pair of bred does. This would give you milk yield following kidding and if they birth bucks, you could harvest for meat or does you could add to your milking flock.

Goat milk also has unlimited potential when it comes to other dairy products; cheeses, kefir, yogurt etc.

PM me if you have any more questions. Good luck!