I've got a really old system (probably around 5 years old) that I'm using as a Unix-based router/firewall/server. It's jet-engine loud, which is ridiculous as it's sitting next to my brand new AMD64 system which is dead silent.

Can a Socket 7 CPU (AMD K6 400) run without a fan? If not, what's a good cooler for it?

Any recommendations for AT (not ATX) power supplies that are quiet?

I had a K6 400 before too. My fan periodically dies however the computer did not die. I think it is possible to run fanless if you have any sort of decent headsink. Honestly my heatsink was pretty dinky however it got burning hot to the touch.

I think any smallish socket 370 or socket A heatsink would able to mount on the lugs around your socket 7. Just be careful not to wear them off.

Oh yeah, I ran a P1-133 fanless, with a stock heatsink not attached by the lugs ('cause one of them broke), without thermal paste, for something like half a year and it worked fine with a small enhancement. Perfectly stable and all. I don't know how long that could have kept on going, as I upgraded. I'm afraid your K6 is a lot hotter, though, so it's probably best not to try that.

As for power supplies, it's luck of the draw really. I had an old non-standard form-factor Compaq PSU with AT plugs and it was a lot quieter than most generic stock PSUs. Failing all else, the computer draws so little power that a fanswap with whatever SPCR's current 80 mm favourite fan is would be perfectly fine. Run it at at 5 V... and enjoy finding that the old hard drive makes noise, too

I had a K6-2 500
I lowered the cpu Voltage to minimum (easy with jumpers), et lowered the fan voltage down to 5v.
It worked just fine and silent (more than the PSU) for 1 or 2 years.

For the PSU, if yours run fine, go for a fan swap, and cut the grills...

I've got a K6-200 with a large all-aluminum socket A heat sink on it. When running on an open bench with no fan or other airflow, it got extremely hot. Although it never did crash or show any other signs of problems. All it would need would be a fan nearby. So if you do find a good PSU, you could probably get away with no fan on the CPU as long as you make sure the the heat sink is in the path of the airflow. Otherwise, just throw a good 80mm fan at 5V near it.

A good AT PSU may be hard to find. Modding an existing one would probably be easiest. I wouldn't expect the power draw to be much in this PC. So you could probably 5V or 7V the fan. That and cutting away any restrictive fan grills (replace with nice wire ones if you still want something) will give you a nice quiet PSU.