It's a Windows 98 PC, and I've backed up every file on the hard drive, so it's obviously lacking in monetary value. I turned it on almost every day from September 1998 to May 2011, though, so it has some sentimental value for me. While expensive back when it was new, it was a well-crafted machine that never had a component fail, despite being used in a hot and (admittedly) dusty environment. A lot of people griped about Windows 98 back in the day, but I always had a good experience with it (except when AOL misbehaved, but that's another story for another day!).

When I set up the PC back in summer, it booted properly, though since the CMOS battery was dead, the BIOS would prompt me for the date and time every time it started. However, it soon grew more finicky - one time it hanged while the operating system was loading, and I had to yank out the plug to get it powered off. From then on, it wouldn't even boot. A few days ago, I bought a new CMOS battery and installed it, hoping that would fix the problem (since some PCs don't even run POST if the battery is completely dead). For good measure, I checked all the connections on the drives and to the motherboard. Yesterday, I hooked up the monitor, keyboard, and mouse, and I turned on the computer. Unfortunately, it didn't even make it to POST. The fan fired up, and the power light and hard disk light stayed on while nothing else happened. This is the same fate that befell my old laptop back in September, and I get the feeling the motherboard's shot.

Is there anything I can try before I throw in the towel?

I'm afraid that this isn't a good sign for your comp.
If it's failing to even make it to POST, it could indicate a failure in the system bios, corruption or worse.

I would check the capacitors, see if any of them are blown.
If even one has been domed out or has leaked, this could indicate a severe problem in the Motherboard.

I'm afraid I don't know how to fix this one.
the lack of a beep code (which has me worried) only makes this harder to diagnose.

Monk's statement posted elsewhere on, I think, "biting the dust."