Some date back to the 1980's http://www.computinghistory.org.uk/det/18689/Compaq-DeskPro-286e-Model-No-2551/

Check your manual for a CMOS reset so it clears what tells it to power up when power is applied then if it powers up it's likely a PSU or motherboard failure. Parts like that are rarely bought new in such old gear. Ebay is where I scrounge those.

But a desktop replacement is too cheap to repair. This really irks owners but others get it.
Example: https://www.groupon.com/deals/gs-hp-compaq-8000-desktop-pc-core-2-duo-windows-10-20-inch-monitor-bundle is 150 and even less with discount codes. In many cases like yours this is an upgrade for the price of repair.