I would suspect the power supply or motherboard.
Absent any sort of testing equipment, the first thing I would do would be look at the fans to see if they're spinning (at least momentarily). Some (a few) motherboards will abort the startup if a fan has failed. I don't know if yours is one of them. If a fan has failed, I would look at whatever component it was cooling as a possible cause for the problem.
If the fans check out okay, I would look closely at the mobo for blown caps (capacitors). There will likely be two types: ceramic capacitors that look like little disks that are
slightly convex on both surfaces, and electrolytic capacitors that look like little cans and usually are mounted upright. Some may be in difficult-to-inspect areas, such as under an overhanging processor cooler.
If any of the ceramic caps are bulging or distorted (bearing in mind that they're slightly convex to begin with), or if the electrolytic caps are leaking or are bulging at the top (they should be flat), then they are blown.
Yes, it's crude and subjective and requires a good eye, especially in the case of the ceramics. It's easier to tell on the electrolytics. But long story short, if you have blown caps, then the motherboard is the problem. The specific problem could be the blown cap(s), or it could be something else on the mobo that cause the cap(s) to blow. Your options then are (1) to find someone who can diagnose the mobo and replace the caps if they're the problem, (2) replace the mobo, or (3) replace the computer.
Note that there are other motherboard-related problems that can occur and which have nothing to do with blown caps. But these would require testing equipment and diagnostics specific to that mobo.
Once the purely visual inspections above are done, you can try re-seating everything, baasically because it's easy and free. (Make sure the machine is unplugged while doing so). It's not unusual for components to creep out of their slots / plugs. But frankly, I don't think this is your issue because when a component that's tested during POST fails or is not detected, it usually triggers an alarm. Remember to observe
anti-static procedures when working on internal computer components. (At the very least, grasp or rest a bare forearm on a metal part of the chassis or case.)
The next thing, assuming you don't have a spare power supply, a voltmeter, or a power supply tester available, would be to pull the CMOS battery out with the power unplugged, leave it like that for a few seconds, replace it, and try to boot up. Sometimes (rarely) something gets screwy in the CMOS settings and it needs to be cleared out. But frankly, it's a long shot. Note that doing this will default all or some of the CMOS settings (and most certainly will reset the clock), which will need to be addressed in BIOS setup once the machine is bootable.
The next thing to check is the power supply, which likely is the problem. You can do this with a power supply tester (preferably), with a voltmeter, or by substituting a known-good power supply. Remember that a power supply should NEVER be plugged in unless there's a load on it (either the motherboard or a dummy load / tester).
If you have a old-fashioned, needle-type voltmeter, set it for DC volts, attach the ground to ground and the hot to the 12V hole in a spare HD plug, with the power supply connected to the mobo. Then try to fire up the machine. If it doesn't read ~12V on the 12V lead, then the power supply is shot.
If it initially reads ~12V, but then drops off more than slightly, then the power supply is shot.
If it bounces around all over the place, or if it reads anything less than ~10V or more than ~14V, then the power supply, and quite possibly the motherboard, are both shot.
If it tests okay, then do the same tests on the 5V side.
If it still tests okay, the power supply may
still be shot. But the above is a quick check using a common tool that you may have laying around the garage somewhere. Using a good power supply tester is more accurate.
The above suggestions aren't comprehensive, but they're a place to start assuming you don't have a lot of diagnostic equipment sitting around your shop.
-Rich