OS X unusably unresponsive after upgrading 10.6 to 10.8
Ultimately I discovered that there exists a Safe Mode, which you can access by pressing and holding Shift after the gray boot screen appears. In this mode the system was responsive again, and I could actually copy my data off.

There also exists a more restrictive Recovery Mode, which you can access by pressing and holding Command+R, and you can use Disk Utility there to copy entire disks around. However, that wasn't necessary for me (until it came time to wipe out and reinstall the OS).