ELKS (Embeddable Linux Kernel Subset) has been around for a while. It's a very stripped down Linux kernel and some apps, like a shell. The big problem with running any good OS on an 8088 is that there is no memory management unit in hardware, so all of that has to be stripped.
People have booted ELKS on things like Tandy 1000s. It would boot on a PCjr with some minor hacking to get around a hole in memory caused by the video buffer.