In my opinion, since you say the Apple system is a desktop, it sounds like a faulty power supply. If that thing is new enough to use an ATX power supply, you're in luck that it will be easy enough to fix. But since the G3 straddled the change at Apple, if it's some older proprietary system, you're going to have to scrounge ebay and craigslist for a replacement.
Same problem for the Compaq, only since that's a laptop, it almost certainly means that the motherboard has to be replaced. If the thing is under warranty, get Compaq to send someone around to do it. Replacing laptop motherboards is fundamentally similar to desktops, but it's often a bit more involved, and is a game of "find the screw that's holding the bloody thing in place."
Could this be causing my machines to not turn on?
I have two machines that I have been using that have this problem...
The Apple G3 M5183 Desktop is working but sometimes when I turn on the power by pressing the button it might not turn on. If I unplug it for a while it works...but why would the power not work to begin with??
(I also remember this problem with other computers, such as an IBM desktop...I don't have this machine anymore...I guess that is good because too many desktops isn't always good if I am worried about them taking up space).
And also I have Compaq SR1053WM. Right now I am still unpacking, don't have ethernet cable, have not set it up to the router, and this Apple desktop is keeping me very busy since I am fixing it and upgrading it already...so because all of this I am taking my time in fixing the Compaq SR1053WM.........Last time I turned it on it would turn off after a few seconds.
Are malfunctioning batteries causing this problem?
And I am wondering, do I need to replace the power supply on the Compaq, since it sounds like a short. The giant power light flickers, the fans reminds me of a car with a loose battery. Or is it the processor or another object could be loose or malfunctioning?