There was a C compiler written in assembly posted on HN 4 years ago [1]!
So yeah, if you wanted to, you could bit-bang an assembler capable of assembling a simple C compiler, then in your simple subset of C you could implement a full C compiler, and from there you can do anything you want!
The grad student approach also sounds interesting. A basic Lisp interpreter can easily fit on a single letter-sized sheet of paper. A single person could hand-compile that, it would just take longer. But, if you're living alone in a cabin in the woods with your own hand-built computer and a personal library of computer books in hard copy, that would be a totally feasible project.
[1] https://news.ycombinator.com/item?id=17851311