Update below, the rest is more or less background information.
After discovering a Genieo install, I vaguely recalled it being, well, not trustworthy. After a quick search I knew I wanted to uninstall it and found this: http://www.thesafemac.com/arg-genieo/. (This guide is actually recommended in another question on this site.) I followed the steps, but made two blunders. I rm-ed the files listed in the guide, thereby skipping the recycle bin (I'm not sure what the English name is). After I finished the list I wanted to make a back-up (second blunder: I should have done that before I tried to uninstall Genieo, my last backup is 76 days old) but noticed that the program needed to unlock my drive didn't open, like all other apps that weren't opened yet. At this point, I wished I had the files still in the recycle bin so I could place them back. Fortunately, I already opened Terminal (I have it still open) (so I can still use command line utilities) and Safari (enabling me to type this question). Safari just crashed. Because I can't launch apps, I assume my computer won't reboot. Please save my iMac.
By the way, I don't notice any obvious traces of Genieo still being present. At least that worked.
I would be happy already if I could just make a backup (but my drive is lockable and I need to launch an app to unlock it, so I can't access my external hard drive) and after that boot from a CD and restoring the old backup or restore from Time Machine (the old backup), in both cases I would have access to my latest files and I could place them back one by one (as restoring the whole snapshot would break the system again).
UPDATE
I had an idea how to fix this: boot from disk, make backup, clean install. Now I just need help with the first and last step (never needed to do that). There is one problem though: my iMac is very old; it came with Tiger, but I upgraded to Snow Leopard with a Mac Box Set. My question now is if I can boot from the CD in the Mac Box Set. The accompanying manual says nothing about this.
Best Answer
Turns out I couldn't access my disk when booting from disc (I apologise to all dyslectics reading this). I discovered, though, that since my last backup (which was 79 days before the crash) I had actually only used the computer very little, so I used a single USB stick to back up all files that were created or edited since then. (These files are fairly easy to find by sorting on last edited date in Finder. Luckily, I had still access to Finder.)
Booting from the disc in the Mac Box Set is indeed possible, I asked at the local Apple Reseller and did it myself. It was very easy (<3 Apple). I just had to press C during startup, see http://support.apple.com/kb/ht1533 for the official list of startup key combinations. (That is where I found the information.) After booting from disc, I used Disk Utility to erase the disk and then reinstalled Snow Leopard. Fixed!