As far as i'm aware, the very first rust compiler was bootstrapped in C and the default rust compiler has since always been written in rust then compiled with its previous version.
above is "i'm pretty sure", but i _think_ the current rust compiler source has instructions on both how to build (with rust) and bootstrap (with C) it.
The rustc binary should run on any modern glibc distro by default and there is builds for other libc libraries too. It's not purely statically linked, it does run against glibc and pulls in a few other dynamic libraries, but most of those should be present if your system can show a TTY with a shell. I'd recommend using rustup to install the newest toolchain (and avoid pulling in an old one from your package manager, rustup can install older toolchains for you and manages it better than your distro with high probability)