Controlling electronic population through chemical doping is one way to tip the balance between competing phases in materials with strong electronic correlations. Vanadium dioxide exhibits a first-order phase transition at around 338 K between a high-temperature, tetragonal, metallic state (T) and a low-temperature, monoclinic, insulating state (M1), driven by electron-electron and electron-lattice interactions. Intercalation of VO2 with atomic hydrogen has been demonstrated, with evidence that this doping suppresses the transition. However, the detailed effects of intercalated H on the crystal and electronic structure of the resulting hydride have not been previously reported. Here we present synchrotron and neutron diffraction studies of this material system, mapping out the structural phase diagram as a function of temperature and hydrogen content. In addition to the original T and M1 phases, we find two orthorhombic phases, O1 and O2, which are stabilized at higher hydrogen content. We present density functional calculations that confirm the metallicity of these states and discuss the physical basis by which hydrogen stabilizes conducting phases, in the context of the metal-insulator transition.