The word which means the same as the group of words "a box or cage for rabbits or small animals" is hutch.

• Lair is a den or resting place of a wild animal.
• Hutch is a box or cage, typically with a wire mesh front, for keeping rabbits or other small domesticated animals.
• Kennel is a small house or shed for a dog or other animal.
• Menagerie is a collection of wild animals kept for exhibition.