Whether you would get evicted depends very much on your landlord and how often they do inspections.
We used have a dog and a cat. The agency and the landlord gave us permission for the dog but not the cat, but never said anything if the cat was around at inspection time.
Gerbals, like hamsters and budgies, are generally no problem if they're kept properly and cages aren't left to get dirty and smelly.
Legally, though, if you haven't asked permission to keep a dog and they discover you've got one, they would be entitled to serve notice.
In your position, I would write a very nice letter to the landlord, care of the letting agency, asking if you might keep a small dog. Explain that you will keep it responsibly and ensure its good behaviour, and that you will take full responsibility for any cleaning or repairs necessitated by said dog.
You may be surprised.