Video card died
Nothing exploded, alas, that would have made for a more gripping story... My iMac's video card just died, and I decided to try and fix this on my own. Unfortunately iFixit didn't have a proper card in stock, so I ordered it elsewhere, but I got the toolkits and a 3 TB HD from iFixit. The HD was a nice upgrade compared to the build in HD (much bigger, and faster) so as I had to take the iMac apart anyway, I went for it, too. The replacement of both HD and video card were incredibly easy and straightforward using iFixit's guides and tools. Fantastic experience, everything done in less than 2 hours (well, it was my first time disassembling an iMac, so I took my time!), and everything working like a charm.
Don't listen to all this noise about iMacs being hard to fix. With iFixit's guides, it's a child's play (ask my 7 year old, he actually helped out!).