Try moth balls. Works for mice. Spread em around where the beaver at. They last a long time, even outside. Do they live in a hole in the ground, like a wood chuck? If so, and they sleep all day, use a smoke bomb to suffocate them while they sleep. Plug the escape hole, Light the bomb, throw it in, and close up the entrance. No stink.

If that don't work, and you can't shoot em, I suggest you put Pam Peaks head on a pike in your yard. Call it a Scare Whore.