Two commands to get you running:
1) rustup target add x86_64-unknown-linux-musl
2) cargo build --target=x86_64-unknown-linux-musl
One thing Go has going for it is that they have a more complete standard library so you can do without the pain of cross-compiling—which is what you're doing if most of your libs assume you're in a glibc environment but you really want musl.
I know because I recently tried compiling a Rust application that had to run on an air-gapped older Debian machine with an old glibc, and the easiest solution was to set up a debian container, download Rust and compile there, instead of fixing all the cargo deps that didn't like to be built with musl.
So I just migrated everything over to that (which consisted of enabling the `rusttls` feature for every crate) and made another build with musl. Everything worked perfectly fine, and since it's not a performance sensitive application, there was basically no drawbacks. The binary became a bit bigger, but still under 4MB (with other assets baked into it) so wasn't a big issue.
Go creates static binaries. I was a bit bummed out to learn that Rust doesn't by default since it requires glibc (and copying a binary to an older distribution fails because glibc is too old).