If you're concerned about issues beyond what Rust's strong safety and type system can protect against (e.g. bugs in those, or in third-party libraries etc.) then you really want separate processes rather than just threads; even if you use seccomp on an untrusted thread, at the OS/CPU level it still has full write access to other threads' memory in the same process.
Either way you'll need to write code designed to run in seccomp carefully (for example allocating extra heap memory might not work) - but the good news is that Rust is a great language for having that control!
There's a reasonably useful discussion on seccomp in Rust which has some suggestions.
The best bet looks like gaol from the Servo project, which is a more general process sandbox (including seccomp). There are also some other lower level seccomp wrappers like this one.
I haven't tried any of this yet, so I'd be interested to hear any other viewpoints/experience.