What about gerbils or hamsters?  Are there any native animals that can be kept as pets?  Like bilbies?  

We had rabbits when we were young ... we had a chinchilla called Cwch (pronounced cootch, it's Welsh for cuddle).  It was hilarious ... every now and then it would escape from its hutch and chase the neighbour's cat around the garden.  It would come into the house sometimes and chew the tv cords ...