If you are talking about were a breed kepts rabbits. It is a rabbitary.
Hutch is answer
Answered
The place is called "port" where ships are kept? woodenboatusa.com
Horses are kept in a stable.
Jean had told her mother that Hazel had kept rabbits.
Hangar
An apiary.
columbarium
Hangar
cote
annathavalam
There is no specific name for a cage for rats, its just based on what the housing is. So a cage for rats is just a cage. (Just like a hutch for rabbits is still a hutch even with no rabbits in)
A nursery is a place where small plants and saplings are kept and grown for transplanting.