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.
Man, I wish I had a do-over. I got rid of my lager fridge, which I reconditioned, a few months back. It was an old fridge. A couple times it lost its cooling and I noticed the clicking. What I did each time was unplug it, wait an hour,plug it back in and it reset and cooled again. Until the last time when it didn't. If I'd thought to look for a site like this, I maybe could've found a relay for it. Like I said, it was old and there might not have been a part for it, but now I'm back to square one. Thanks for the link!