ELKS v0.8.1 by Gregory Haerr is a version of Linux for the processors 8086, 8088, 80188, 80186, 80286, NEC V20 and V30 and compatibles. ELKS can work with 256 Kb of RAM but you will be better with 512 Kb (128 Kb only with a ROM). It doesn't need a memory management unit (MMU) which you get on more recent processors.

You can see the video below of ELKS running on an Amstrad PC 2086. You will find more informations on the wiki and the documentation of ELKS. And yes Doom is running under ELKS. You can even test ELKS online.