It starts working again a couple of times, but freezes when I try to insert a cat5e into the back...Other than that it's fine for several hours. Then it just freezes after like 10-15 minutes each time.
My guess is the PSU or RAM. Test the memory itself (after disconnecting unnecssary things like fans, NICs, HDDs)?
What OS? If it's Vista or W7, use the "extended" memory diagnostic test, that starts when you boot from the installation DVD. If it's linux, MemTest86+ is on installation disks (at least for Ubuntu and OpenSUSE distributions) If the test fails, try each stick of RAM independently.
My bet is the PSU, but running this test should help support that theory.
When you were moving places and rearranging the furniture, is there a chance you moved the computer case while the HDDs were spinning?