My kegerator stopped working last week. It wouldn't chill and the only noise it made was a loud click every few minutes. This is exactly what happened to my last chest freezer that died, which I replaced. This was a lot worse, though, since I have the fridge built into a larger bar and replacing it would be a total PITA.
I decided to do some more research and it sounded like it could just be the start relay. I found the right part and ordered from repairclinic.com, and lo and behold, it works again. So if your fridge/freezer won't start and you don't hear any noise from the compressor, you might be able to fix it for cheap instead of throwing it out.