I called Sony support and they advised a safe mode repair followed by a rebuild, and now the machine has got itself in a rebuild cycle where it switches off before completing it each time. It is now in a completely failed state and I am without doubt that the root cause lies with the stress on the hardware caused by Skyrim.
Similar thing happened to mine. One time Skyrim froze and crashed mid game out of nowhere, and upon restart it got stuck in a loop at the rebuild file system screen and kept restarting itself over and over. Going into safe mode and recovering the database or restoring the system at that point didn't help.
What helped was to wipe the drive on my computer, install the PS3 4.0 firmware to a USB thumbdrive, put the wiped drive back into the PS3 and run a clean install of the system. After that, the system worked fine, and is acting like normal with all other games. Maybe give this a try before shipping it out. If it still fails with other games afterwards, then send it in for repair or have someone look at it. I've been monitoring and using my system a lot since, and it is acting normally, with no indication of a deeper issue.
Could also try replacing the hard drive altogether. It needs a 2.5" Sata drive, either 5400 or 7200 rpm, or an SSD. Might be worth it if out of warranty and the repair cost is high, yet you want to keep the system rather than get a new one.
If it is a deeper issue, like the thermal paste or the power supply, then repair or replace are the only options. I'm simply offering a suggestion which worked for me, in case it might work for you. It may not.
Some notes... any protected PSN save games are probably lost and unrecoverable (which. really. svcks). After reinstalling and signing in with the PSN, go to Account Management > System Activation to reactivate your system for licenses; and under Transaction Management > Downloads, there will be a list of all items puchased from the PSN, which can be downloaded and installed again.
Hope this helps. Good luck with it. I feel with you, and wish you didn't have to deal with this.