This sounds like the power supply to me, but RAM is a possibility as well.
Many shops will test the power supply for free. The last time I tried to
check a Compaq power supply I was very annoyed to find it had a proprietary
connector. It would be luck if you had an existing PS that would fit.
Most power supplies can be found with a Google search (Get the model number
off the PS) and replacement is a fairly easy job.
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The computer freezes or locks up (I don't know the proper computer term
for this) and then nothing will work, no cursor movement, no nothing,
even Cntrl-Alt-Del doesn't do anything and the only solution is to power
down, wait ten or so seconds, and then power up.  Sometimes that works
and sometimes (many more times) it doesn't.  I can try this many times
and maybe, just maybe, it'll work on the 10th or so try.  If it doesn't,
then I'll go off, come back and try again.  After some time, with
repeated attempts I'll get it up and can use it for a period of time
varying from minutes to days - literally.  I leave the computer "on" as I
let it do searches (when I'm not using it) for the SETI program (I teach
Astronomy at a Jr. college here).  It'll often run OK overnight, but
sometimes it'll freeze after a few minutes - even after a new boot up.

Other symptoms:  When I turn the power back on after a freeze up, I can
tell instantly if it's gonna boot or not.  Two things:tell me this:  The
"A" drive comes on and its light blinks if it's gonna boot, but doesn't
come on at all if it's a dead machine & the monitor light turns from
orange to green if its gonna boot, but stays orange if it's still dead.
These indicator lights are almost instantaneous - - long before the
machine has had time to really 'do anything'.

My (fuzzy) thoughts:  Since, after it died, but starts at a later time, I
thought it might be temperature problem.  I don't think this is so - I've
opened the case, removed (with many misgivings), and thoroughly cleaned
the processor chip fan and, etc.  The other evidence is that even after
'resting' overnight occasionally it won't boot - surely things would have
cooled down by then.

More (fuzzy) thoughts: Can my power supply be the culprit here?  I just
can't see a temperature problem, or Virus, spyware, adware problem.  The
PS is a 250 watt unit, but I've added nothing to the machine except a 100
MHz Omega Zip drive - that's worked perfectly for years.

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