Linux clustering would definitely be the way to go. Just realize that these systems will have to be dedicated to running server software. Because of the speed of the processors you will not be able to run current desktop environments, OS emulation, or games of any type. If you stick to purely running shells and the limited amount of protocols you described it should work ok.

You do not get any benefits by having them in the same case. You cannot share power supplies as most 386 power supplies are not going to power multiple motherboards. You do not need to worry about a KVS appliance or sharing a monitor as you would only be shelling to the different systems you could use SSH to control the other members of the cluster. You would just need to have a video card/monitor installed at one point to do the initial configuration.

Sounds like a fun project. Just realize that it will not be a simple thing to get it all up and running if you are not at all experienced with clustering software. The information is out there, you will just need to spend some time reading before you start to install all the software.