I suppose the right answer is "it depends". For most unikernels the answer is no: they are highly specialized, often targeting a single application or language, and built on top of small/basic OSes. Some of them have targeted various levels of POSIX compatibility, e.g., OSv (http://osv.io/) and HermiTux (https://ssrg-vt.github.io/hermitux/) though they don't always support a large number of syscalls/applications. An older project was Rump (largely abandoned), which used the NetBSD kernel to be (mostly) POSIX compatible, though performance wasn't great and it's largely abandoned. An active project targeting at least partial POSIX compatibility is Unikraft (http://www.unikraft.org/).
So can apps using a RustyHermit use the std lib, or should all be no_std?