Well, I learned today that the 3-year "Red Ring of Death" warranty on the XBox360 has expired. So, there wasn't any reason that I shouldn't at least try to fix the problem myself, if it was indeed repairable. I read up a bit on what the typical issues are that cause the problem, and it seems that it is almost always related to the the GPU. Either its overheating or there are some cold solder joints.
In any case, in order to fix the potential issues, you've got to open up the unit and do things to it. The fix that I am trying is the "X-Clamp" fix, where you end up replacing these X-clamp things that hold the heatsinks in place. Part way through the step-by-step repair instructions that I was reading, I realized that I needed some stuff that I didn't have. So, tomorrow I'm going to have to buy some washers, thermal paste, and thermal paste remover.
Here are a few pictures of the current state of my piece of crap XBox 360.