If you've got an apple laptop, you can mount it in target disk mode over firewire How to use FireWire target disk mode
Then you can verify all the data is backed up, and then take it to an apple store, if it's still under warranty, or put it on a new system. This is the first time I've heard of a problem like this for the minis, so I'm guessing fan failure is probably the right call, and hopefully nothing was cooked too bad. My laptop sometimes overheats, and when it does, it shuts off before any damage happens.