I have an ancient 386 with an on/off switch on the PSU. It just died. Power doesn't come on any more. I have another PSU with correct cables but no switch. Where do you switch an old 386 PSU?
In that era, most clones used a big latching (non-momentary, either push button or toggle)
DPST switch on the computer case that physically connected the power in socket on the PSU to the rest of the PSU (all power for the computer actually flowed through the switch), attached to the PSU with a dedicated cable with spade terminals. A PSU of that era without a switch on the PSU should have that cable, and if you don't hook a switch up to the switch cable you won't have any way to turn on the PSU.
You should be able to use any latching DPST switch with the right sized terminals that's rated for mains voltage and the current your PSU draws (should be less than 5A, since PSUs of those days were generally no more than 300W). Finding a place to mount that switch is another question, of course.
You can also get adapter harnesses that let you plug an ATX PSU into an AT format motherboard (again, you're on your own for mounting), but the potential drawback to those is that modern ATX power supplies deliver most power on the 12V rail and not a lot on the 5V rail, but old AT format machines drew most of their motherboard power from the 5V rail. You'll need to read the specs for both the old AT PSU and the new ATX PSU to make sure it'll actually deliver enough +5V.
Or what would break on an old switched PSU?
This is a bit pedantic, but it may help if you're trying to find info to fix the old PSU. "Switched PSU" usually implies
switched-mode PSU, but in the 386 days plenty of PSUs were the older (and less efficient) linear power supplies that used a big, heavy transformer.
386 era is what, early 90s.
Mid to late 80s, actually, so more like almost 40 years old. The 486 came out in 1989, and the Pentium came out in 1993.
ATX came out right around when the Pentium Pro did in 1995, but didn't really become the standard until the Pentium II days.
If a replacement PSU doesn't have a switch, you could always hook it to a surge protector and use the switch on that to power on and off.
As noted above, this is very unlikely to work. The PSU power switches back then actually physically connected and disconnected the mains hot and neutral wires, so without a switch in place and turned on, there's no connection between the PSU's AC input socket and the rest of PSU.