Normally I would just say this sounds like a defective PSU.
However I have, within the last year, come across a suspiciously high number of eMachines systems a couple of years old with a Socket 478 motherboard (Intel Celeron or P4 processor) where the motherboard has been taken out too.
My advice would be to buy a new PSU, they're only cheap. Remove the old PSU and connect the wires of the new one up to the motherboard. If you get no display, no beeps, chances are the motherboard is dead too.
If you can still get a Socket 478 motherboard (probably micro ATX rather than full size ATX if I remember the case size correctly) then buy it, fit it, put the new PSU in and job done. Try to buy a new board that has the same chipset as the original, be it VIA, Intel, or SiS etc, to try and eliminate the risk of having to re-install Windows XP. You will then need to load all the drivers for the new motherboard (on-board sound, network, etc).
If you don't know what you're doing or lack confidence maybe a more experienced friend could help you out.
As a DIY repair this would be quite cost effective. But if you have to take it to a repair workshop to get it fixed, the total bill including labour and parts may make the repair uneconomical depending on the age and specification of the machine.
Hope this helps.