  We analyze $L^2$-normalized solutions of nonlinear Schr\"odinger systems of
Gross-Pitaevskii type, on bounded domains, with homogeneous Dirichlet boundary
conditions. We provide sufficient conditions for the existence of orbitally
stable standing waves. Such waves correspond to global minimizers of the
associated energy in the $L^2$-subcritical and critical cases, and to local
ones in the $L^2$-supercritical case. Notably, our study includes also the
Sobolev-critical case.
