Is there a process for making a fully static binary (say if I were to substitute an alternate libc, or provide my own low level wrappers for syscalls, or whatnot)?

Yes, that's totally possible. You can write a kernel in Rust if you want.

...is it documented anywhere? where?

http://doc.rust-lang.org/book/unsafe.html#avoiding-the-stand... is the closest we have, currently. https://github.com/rust-lang/rust/wiki/Operating-system-deve... has a bunch of stuff, and https://github.com/charliesome/rustboot and its forks are examples that I know of.

The example minimal do-nothing program there is much smaller (8k), but still results in a dynamic binary requiring glibc.

The only unresolved symbol is: __libc_start_main@@GLIBC_2.2.5

I believe musl supports static linking

Any reason one couldn't use musl instead for static linking? (http://www.musl-libc.org/)

and for many, there are legal issues with statically linking glibc due to its license.