I found a working answer on Server Fault from reddit.
Even though I did disable blanking, the screen still goes blank, but I can bring it back with a key press. I can live with that for now.
The trick was to install `console-tools` and configure it from `/etc/console-tools/config` to have
BLANK_TIME=0
POWERDOWN_TIME=0
Added on 2013-11-15: After an update, this broke down again! The display would blank and never unblank. This time I had to update the file in `/etc/kbd/config`. After fixing the file the computer remained on all night without blanking the display.