That is a Jack Russel Terrier, the greatest dog to ever dog. ~12 lbs of pure energy that can be sated only with digging and 'the chase.'

To keep that little bundle of chaos indoors full time is to invite destruction itself into your home.

A mixed indoor-outdoor setup is ideal, but if any dog is going to thrive in an outdoor environment (fenced with shelter, water, food, and some additional protection under the fence to discourage tunneling out), it would be that beautiful creature right there.