Inferno runs on stock Nintendo DS Lite with 4 MB RAM, but it can run on even smaller hardware.
https://en.wikipedia.org/wiki/Inferno_os

There's a cut-down version of Linux that runs on the RC2014 Z80 SBC, which IIRC has only 512 KB of RAM. Demonstration here:
https://youtube.com/watch?v=1WG8zopGzaA
Watch his other videos too, they're pretty nice. I'm just not that crazy about Linux anymore though, and run OpenBSD on my laptop, but I'm not even too crazy about OpenBSD. It's gotten quite bloated as well. It used to run fine on Motorola 68030 systems (like Amiga, Atari ST, old Macs, etc.), but they abandonned those machines years ago. Anyway for the RC2014 I think CP/M makes the most sense, and you could probably code a TCP/IP stack for that. After all, they managed to do it with a Commodore 64 that has much less memory:
https://en.wikipedia.org/wiki/LUnix

Here's another guy with some interesting videos about making custom computers with old processors and not much memory.
https://youtube.com/watch?v=BP7sjnVzTqw

BTW CP/M itself will run fine on an 8080 or Z80 computer with just 64 KB RAM. The RC2014 is really souped-up compared to the stuff from the 80's.

A stock Amiga 1000 has only 256 KB RAM but still runs a full multitasking OS with GUI. Ditto with the original Atari ST. I consider them to be the last of the interesting machines. There was much more impressive stuff that came after (SGI Indigo, etc.) but nothing as tightly designed and affordable.