The server has been running for ~48 hours with no problems, so I'm willing to declare this problem solved. In case anyone comes across this problem in the future, setting "binary-type = hex" in abe.conf did the trick.
So Abe on Postgre won't work with default Abe config? Maybe we should do some workaround in Electrum and expect binary data there. Converting binary data to hex isn't so hard and I bet that storing binary data in database will save lot of storage space...