Your machine should have been able to handle the upgrade. But, maybe there is a problem with either software on the disk, or the disk itself (or both). Assuming you get things back working, make sure to start making backups often!
I also have a Mac Mini, but it is a later model. I actually replaced the slow, 5400 rpm 1 TB internal drive with a 256 gig SSD, and I had no issues upgrading from Mavericks to Yosemite.
In any event, your mini has a Firewire 800 port. If you have another Mac that has either a Firewire 800 port, or a Thunderbolt port (you would then need a Thunderbolt-to-Firewire 800 Adapter), you could boot your mini in Target Disk Mode, and use whatever disk repair software you have on the other "good" Mac to try and resolve your issue. If you don't have such another Mac, then your other option would be to attach a DVD drive to your mini, insert the OS 10.6.8 DVD into that drive (hopefully you still have it), and boot the mini from the DVD. You could then use Disk Utility on that DVD to try and "fix" the issue on your drive.
In terms of "fixing" things, if the OS "stuff" is problematic, hopefully you could recover the other files/folders on the drive, and copy them somewhere else. Then, you could erase the drive, format and partition it, and then do a direct installation of Yosemite. Also, the version of Yosemite is now up to 10.10.3. It would be best if you upgraded to that version, after getting 10.10 onto the mini.