  Hierarchically hyperbolic spaces (HHSs) are a large class of spaces that
provide a unified framework for studying the mapping class group, right-angled
Artin and Coxeter groups, and many 3--manifold groups. We investigate strongly
quasiconvex subsets in this class and characterize them in terms of their
contracting properties, relative divergence, the coarse median structure, and
the hierarchical structure itself. Along the way, we obtain new tools to study
HHSs, including two new equivalent definitions of hierarchical quasiconvexity
and a version of the bounded geodesic image property for strongly quasconvex
subsets. Utilizing our characterization, we prove that the hyperbolically
embedded subgroups of hierarchically hyperbolic groups are precisely those
which are almost malnormal and strongly quasiconvex, producing a new result in
the case of the mapping class group. We also apply our characterization to
study strongly quasiconvex subsets in several specific examples of HHSs. We
show that while many commonly studied HHSs have the property that that every
strongly quasiconvex subset is either hyperbolic or coarsely covers the entire
space, right-angled Coxeter groups exhibit a wide variety of strongly
quasiconvex subsets.
