The solution to the problem of the non-booting computer either had to do with two bad RAM modules or two bad RAM slots (and thus a very unhealthy, yet miraculously still working, motherboard). Sometime later I will determine whether it's a RAM stick or a slot/motherboard problem, but I'm glad the old computer is working again. (The installed Firewire card doesn't work either, so this might be a sign of an ailing motherboard.)
In any case, replacing the power supply unit (PSU) with a new one from CPUtopia, in my opinion, contributed to the ultimate diagnosis and solution. Replacing the hard drive's PCB (see Part One) cost about $40, and replacing the computer's PSU cost about $25. In the end, along with partly dealing with the RAM issue, I retrieved some irreplaceable data for about $65 in supplies (and a priceless amount of effort, research, experience, elbow grease, trial, and error).
Right now, I'm getting situated on a new machine. It's taking some time to make it like my trusty (now ailing) computer, but it's getting there -- and hopefully this setup will be an improvement. Okay, it is an improvement. I'm using my studio monitors to listen to iTunes mp3s, instead of using some tinny and tiny computer speakers -- so it's fantastic already!
So -- if you don't want to strain your eyes to read my poor penmanship -- the following non-exhaustive list could be the cause of a non-working, yet salvageable, computer:
1. Hard drive with bad master boot record (MBR). Use another computer to fix that hard drive. You'll need some special hard drive data and power cables.
2. Hard drive with fried printed circuit board (PCB). Replace it with an almost-exact PCB (chip, firmware, etc.).
3. Bad power supply unit (PSU). Replace it with a new one, compatible with your computer.
4. Bad RAM. Remove the bad module(s), and deal with a slower machine for now, or until you can get some good RAM modules. Hopefully the RAM expansion slots are still good. Otherwise the problem might be a...
5. Bad motherboard. If the slots are bad, don't install stuff in those slots. (That's what she...okay, okay, okay.) If things still don't work, you might have to replace the motherboard. You'd have to seek help from a professional computer nerd or something.
6. Some other issue...???
Now it could be possible to just take out the hard drive, and grab the files using another computer. You'd have to get some converter cables to plug one end to the hard drive and another to a USB port. If you need to deauthorize some applications, though, you'd have to do some fancy tricks to boot that hard drive somewhere else. Do some research, ask for some help (if necessary), and good luck!
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