Identifying the problem is probably the hardest part. I'm studying electrical engineering, and my desktop motherboard refused to boot randomly one day. I carefully inspected it, but there were no components with any visible damage (blown capacitor or something like that). Luckily, it was under warranty, so I mailed it in to have it repaired. When they sent it back, I inspected it again, and I was able to determine that it was the exact same board (based on the pattern of dust and smudges), but I couldn't find where they had done the repair. I couldn't find even a trace of hand soldering.
My best guess is that it was a BIOS or power regulation problem, based on the complete deadness of the board. A BIOS chip isn't something you can replace yourself since I would guess most BIOS chips are custom-made by BIOS manufacturers or made for a specific motherboard. The motherboard manufacturer probably isn't going to sell you a BIOS chip. For power regulation, the board is so complicated that I wouldn't even know where to begin diagnosing it. The PCB is 4 or 8 layers, so it is almost impossible to figure out where every trace goes, and it would be almost impossible to understand everything without detailed documentation (which the motherboard manufacturer certainly will not provide-- that would let you make your own knockoffs of their product). Even with documentation, it would probably take someone unfamiliar with the design many hours of probing to find the problem.
Some stuff is repairable by soldering, though. I recently fried a PC power supply by trying to draw too much current from it. I took it apart, and there was one clearly burned trace on the PCB. I bridged it with some wire and the power supply works just fine now. That won't always work, of course. I fried another one, and it had a similar burned trace on the bottom, but when I bridged it with wire, the power supply refused to work properly or put out a stable voltage. I think it was covered under warranty, though.
I have also fixed my iPod Touch when the headphone hack broke. It started playing only karaoke, which happens when one of the contacts in the headphone jack breaks and it only plays the difference between the stereo channels. After a lot of prying to get it open, I soldered in a new headphone jack, and I dremeled a hole in the back case for the cord to come out the bottom.
https://forums.macrumors.com/threads/735778/
I think Steve would have a heart attack if he saw my iPod.
My cellphone's USB jack is starting to get loose from too much use, and it sometimes doesn't charge unless I wiggle the cable. When it gets worse, I plan on putting in a new USB jack, hopefully not sticking out the side if I can find one the same form factor as the original. It might be tough to take apart, though, since the screws are Torx T2 or something smaller than any tools I have.