Can I save this computer? (or at least some of the data)

I have had this iMac since about 2008. The only thing that I've changed in terms of hardware is a RAM upgrade to 4GB. Ideally, I would like to continue using it, but at the minimum, I want to recover some files on it which were not backed up on any other drive.

On Sunday of last week the computer suddenly displayed the dreaded 'prohibition' or circle-with-cross symbol at startup. I tried several things to get it to boot properly including a PRAM reset, SMC reset, and booting into Safe Mode. None of this was successful - after I reset the PRAM, the machine would automatically reboot in a 5-minute cycle. I unplugged it and walked away to evaluate my other options.

Two days later, I decided to turn on the computer again and was delighted to be able to log in! It seemed that the problem had magically disappeared. I immediately went to work to try to back up some files onto an external HD. However, the computer was running very slow due to Spotlight indexing the machine. I ran a terminal command to kill Spotlight, but, the computer had slowed to a crawl. So I rebooted.

On my second successful log in I rushed to copy my documents to the external HD, but I was blocked by an "error code -36" ... The Finder could not copy the file because the ".DS_Store" file could not be read or written. Again, I tried to use Terminal to delete all the .DS_Store files, but the computer had ground to a halt about 20 minutes into the process. I had to shut it down.

I have not been able to log in successfully since then. Sometimes the computer will get to the log in screen but nothing seems to happen when I attempt to log in. Other times it just hangs.

My suspicion is that there is a problem with the hard drive. I think it may be overheating or seizing up - this would seem to explain why I cannot keep the computer running for more than about 20 minutes. The computer is not making any unusual noises or smells.

What can I do to revive this machine or recover some of the files?