Crazy question: Is it possible that PC was retired because it broke? Is it possible someone had already gone in and mucked things up trying to fix it, or was it known to work the last time it was used and left intact until you received it?
Do you see the entire sequence to boot, something like turn it on, (maybe) get a beep, a few seconds spent on bios and querying HDD, then rapid succession of HDD activity LED blinks like it's trying to boot, then that activity markedly decreases suggesting it has finished booting?
If it seems to do all that, try a different video card. If there was a lot of gunk inside, and it's a separate video card (not integrated video), you might also try pulling the card out and spraying the motherboard slot with contact cleaner, and wiping the card contacts.
If it doesn't do all that, look at where it stops. Have you tried clearing CMOS? Some systems of that era didn't halt-idle right if at all and could be hard on the electrolytic capacitors so I'd examine the board, and then in the PSU for bad ones, or of course if the board looks good, try a different PSU.
In "rare" cases I have come across systems that won't even POST if the battery is drained, so you might try a new battery or at least measure voltage on the one in it. I couldn't tell you the exact threshold of voltage being too low on a CR2032 to cause a problem but if below 2.8V I'd definitely replace it. Considering the age of the system, it's due for a new battery anyway.
