Old PC Occasionally Won't Start, Doesn't Power Down all the time

Old PC Occasionally Won't Start, Doesn't Power Down all the time
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I have several computers, but I am having a problem with an old computer I
keep around for testing/experimenting purposes (trying out new OS's, and
generally tinkering with).

This computer is an old gateway g6-300, Pentium II 300, 128MB ram, 8.4
maxtor hdd

A couple of weeks ago I hit the power button and nothing happened. I hit it
again and again. nothing. I was getting ready to leave to do some errands
and didn't have time to fool with it, so I left it for the time being.

A day or two later I went to start it and it started no problem. It
currently has win98se with most of the MS win98 updates installed. I also
noticed if I choose "restart" for the shut down menu, it acts like it's
going to shut down, you see the win 98 "your computer is shutting down."
screen, the screen goes black, but the computer doesn't shut down (lights
are still on and I hear the fans going).

However if I choose "shut down" from the shut down menu, it does actually
power down all the way.

Last night it also did a strange thing where after I pressed the power
button, it began to power up and then shut itself off after about 4-6
seconds. I hit the power button again and it did the same thing. Again and
it did it a third time. I un plugged in and plugged it back in and hit the
power button again and suddenly it powered on normally.

I am guessing perhaps the power supply is starting to go, and I am wondering
if anyone out there has experienced a total or partial power supply failure.
I have never actually had a power supply go on me before. Does this sound
like what might be happening? I don't have another power supply lying around
to swap out and see if it makes a difference. I also don't have a multimeter
or any equipment to test the power supply on hand ( I guess I know what I
may need to get if I really want to get to the bottom of this).

Thoughts?

-- A