(To start with the obvious: you have checked that it isn't just a fuse that has blown somewhere, haven't you? I do not say this in judgement, I once spent an entire day taking a whole computer to bits and doing careful troubleshooting only to find that the fuse had blown, and I could have verified this in about ten seconds)
Depends on what the failure is. If it actually is the power supply you're actually pretty lucky.
Power supplies are relatively straightforward, because they are to an extent replaceable even if you have to do a bit of work to do so.
Personally, given that I'm not trying to keep most of my stuff in any kind of "museum condition", I tend to replace PSUs with modern ones, adapting the pinout as necessary, and being careful about electrical safety on the mains side. This sounds like a pain but it is by far the less stressful and dangerous approach.
Again for power supplies, there are specialists who will repair them - they're not general purpose electronics repair places, but switch-mode power supplies are so common that there are people who work on them. That said, this is rarely cost-effective unless there's something weird and exotic about the original PSU (or you're deliberately trying to keep the machine "stock").
I'm very cautious about working on power supplies beyond recapping myself. The risk everyone talks about is electrocution, and this is a real risk, but the thing I'm actually worried about mostly is fire, especially given that insurance contracts generally look askance at people doing stupid stuff with mains. I'm not at all convinced I'm good enough to get stuff right. This caution is especially necessary when one considers that the machine will at some point leave one's custody and go to a new owner, and IMO it behooves one to not put unnecessary electrical safety risks on future owners.
I've had conversations here which go along the lines of "it's my machine and I'll do what I like to it" or "screw the next owner, got mine", which I think is a little alarming when it comes to electrical safety. Caveat emptor is a useful principle, especially for second-hand goods, but personally I don't think I'd like burning down another forumer's house much more than I'd like burning down my own. Perhaps I'm just being one of those weak softies that it appears to be fashionable to rant about at the moment, though.
But do verify that it's the PSU that's actually gone before you spend time and money attacking it. I agree that the 5V standby being missing is pretty compelling evidence here, though.
