Is it technically possible to run z/OS, or some interesting approximation thereof, on a "normal" x86 box?

Configurations that would be unusable in any sort of production scenario would probably remain interesting.

Yes but not legal checkout MVS Tur(n)key:

http://wotho.ethz.ch/tk4-/

And run it with the new Hercules emulator:

https://github.com/SDL-Hercules-390/hyperion

MVS is near identical to z/OS and the Turnkey Distro has much more compilers and tools..and games already installed.