I bought a PS/2 with the intent of robbing it's case for a restomod project, but since that is on the backburner right now I decided to see if it would work. When I flip the power switch, the PSU comes on and so do the disk drives. No video or audio is outputted from the machine, though. I checked the pins going to the disk drives with a multimeter and fount they are operating at a solid 5.5 volts DC. Motherboard cables are good as well. Any help would be appreciated!
Remove any cards, floppy drives, etc. that are unnecessary for showing the power-on screen or hearing the POST beep, then try to power on. If that still doesn't work, inspect the motherboard for bulging or leaking capacitors, leaky battery, or other obvious damage. Clean out any dust, as it may cause short circuiting. Reseat any socketed chips and any connectors. Check connectors for corrosion. Power on and check if any chips get unusually hot.
If this fails, find a second identical but working IBM PS/2 Model 53, and swap parts until you narrow down the culprit.
traal's answer provides some good ideas. I'd like to add a few.
First, the PS/2 (at least the one I had back in the day, first half of the 1990s some time) will beep and do a RAM check on power up during POST, then proceed to try to boot the operating system. If I recall correctly, it will do so in that order, but it should definitely try to beep at you during the BIOS POST or bootstrap process if things are good.
On the motherboard, find the connector for the internal speaker (or find the relevant locations on the speaker itself), and plug in some way to measure the voltage across those two pins. An oscilloscope with a long hold time would be great, but I suspect a simple multimeter (set to measure AC volts) or voltmeter could do in a pinch. See if there's a waveform there; if there is, then the computer is trying to beep at you but the speaker is somehow faulty; likely a good sign, because it should mean that the CPU is up and running. If you are measuring at the speaker, make sure the cables themselves are good by measuring their resistance with the computer powered off; the resistance should be very close to zero. If the computer is trying to beep at you, you should be seeing an AC waveform at a few hundred Hz and maybe a handful of volts peak-to-peak here for the duration of the beep, and zero volts otherwise.
Second, if it appears that the CPU is not powering on but there's DC present from the PSU to other peripherals (as your troubleshooting suggests), I would want to verify the "power good" signal from the PSU. I can't readily locate anything on what motherboard form factor and particularly motherboard power connector the PS/2 used (I might try again later), but if it's the same power connector as AT or Baby AT, then it would likely (but not certainly!) be pin 8.1 to pin 8.5 or 8.6 (power good to ground). While applying a normal load, check the voltage between those pins to make sure it is decidedly non-zero DC and holding steady. The CPU won't power on before the PSU indicates that the power is "good", but other peripherals might if they don't particularly care about the purity of the DC, such as drive motors, or even don't have direct access to the power good signal. If this is the problem, then it seems likely that the PSU is somehow at fault; either it is not providing good enough DC to be happy about itself, or it is but the "power good" detection circuitry is malfunctioning somehow. At that point, unless you can source a known good spare, you are probably looking at opening the PSU and poking around inside it.