Well, it took longer to sort out than it should have.
The server had locked up and wasn't responding even to ping. No
joy on the precautionary power cycle. So I hooked up a monitor.

Well that's not good.
But nothing had been moved recently (unless someone had been doing
server room parkour), so I hauled it home to take a look. Mucking
about with different kernels would eventually let it run long enough
(I think by coincidence really, but rescue mode helped) to get me:

which, aha, is ECCRAM error reports. All in the same bank.
Not that the banks are labelled, of course. And unless the system is
stressed hard it may not trigger the problem for a while.
But after a few tries I was able to eliminate half the RAM as
"potentially dodgy" and put the machine back into service on the other
half.
This is a box which has to be running as much as possible, but I'll
memtest the old SIMMs at home to see if I can pin it down. Since this
machine is old enough to run on DDR2, though, replacements aren't easy
to find any more; and there's at least a 50% chance it's a mainboard
problem anyway. It'll probably be better to replace it. Anyone know a
good builder of 2U servers? (In the UK.)
Tired now. (It's much easier for me to drive into London late at night
than to move the thing by public transport, and I ended up doing both.)