iMac sounds like it's starting up then shutting down immediately I have an iMac. Since yesterday, it has been making sounds where it seems like a hard drive is starting up then shutting down immediately. However, that can't be possible of course because the computer technically works fine—the sound occurs while I'm using the computer, and it doesn't lock up or anything like that. (Also, I thought it was an external HD but I put my ear to the back of the iMac and it is definitely coming from there.)

This started happening around the same time that I started watching high definition videos (720p h264) of a TV series. I had watched maybe ten episodes before the sounds started occurring. That would be perhaps ten hours over a 20 hour span. Also, the computer was up (hadn't shut down) for 72 hours straight.

I'm thinking it could be a fan problem perhaps, or a power supply problem?

These sounds have never happened before, even when doing stuff like playing high performance computer games.

I have also noticed that there is a high-pitched buzzing sound that occurs when this startup/shutdown sound happens. Also, the startup/shutdown sound seems to happen consistently when I start up my computer, so perhaps it IS a fan or power supply related problem?
Open it up, check for dust, reinstall the Mac OS X (fresh installation), Check whether the fan is functioning properly. If all these doesn't solve the issue, then probably you will need to get it examined by an Apple service guy.