After a few (read:lots) more crashes, I've found out a few things.
First of all, the problem isn't really that the screen goes blank, it's clearly a full system crash. Guess I'm just used to crashes being a bit more...visual. Doesn't linux have a BSOD or some such?
Second, the cpu shouldn't be a problem. It's clearly underpowered, but not so much that it should cause crashes (load often reached 100%, but never stayed there for long periods of time, no matter what I was doing). For comparison, the computer's run Win2k since ancient history, without any real performance or stability issues (until it was overrun by viruses and trojans).
Is there some kind of event log I could check for any weird stuff around the crashes?
Is there any way I could make the system lighter (less processor load, for example)?
And does anyone have a good temperature monitoring app? I doubt it's overheating, but could be worth checking...