I'd really like for that to happen (compiling to a standalone binary is one of the reasons I picked Rust), but I have no idea how to go about doing it...
You could just distribute the executable you get when you run `cargo build`. rustc / cargo statically link the Rust standard library into the app, so it should work fine on a machine without Rust installed. It'll still dynamically link C libraries.
One catch is that (especially if you're using C libraries other than libc, but sometimes with libc) you'll need to worry a little bit about forwards- and backwards-compatibility of those libraries. It's somewhat safer to find a semi-old distro like Ubuntu 12.04, build there, and see if it works everywhere newer.
This is kind of a pain, and the long-term answer is to get Rust into the distros so that they can build software written in Rust. (This doesn't require rustc being installed on any machine other than the builders.) In the short term, a cross-distro package generator for binary crates sounds like a maybe-fun project....
Basically every system has libc on it already, so it's not a big deal. Unless you build on a new system, and then distribute to an old one. Most people pick which systems they'd like to target, and build against the lowest available version.
