Some more information/tests:
I disconnected the hard disk and booted into the BIOS. I watched the real time monitoring of temperature and voltages until the board switched off again - I didn't see any suspicious. Temp was well below 30 degree celsius (86 F).
Without the hard disk I noticed a high pitched sound coming from the board (it's not the PSU because the chassis has an external one). I tried again - the noise is definitely there.
One more observation. When I try to turn on the computer right after it shut down it will not start. When I pull the plug from the PSU and wait some time it will start again (but always shutting off after a few minutes).