It takes some custom work, but, YES, Linux can still run on an old 386, There is a custom library that you will have to include and compile the kernel. Modern kernels are compiled for newer processors. You might have to look up an older version of the compiler, but that is easily found online.

If you do try this, please Please PLEASE don't use a modern desktop, as it might take a half hour for the mouse cursor to move. You want a very light desktop. TWM would be a good choice.

That said, a 386 might make a good router or a dedicated server for many applications. 20 MHZ is enough speed for a surprising number of uses.