During many of its applications (especially as a catalyst support material), MoO3 acts as a medium for hydrogen storage via hydrogen spillover (H atom donation from proton and electron sources to a support), for which the energetics of H atoms on MoO3 are of importance. Despite the seeming simplicity of hydrogen spillover, previously reported ab initio results for the H adsorption on MoO3 contradict both experimental work and other ab initio results. In the present study, we resolve these discrepancies and provide a comprehensive ab initio understanding of H adsorption for MoO3 in the bulk and on the surface. To this end, we systematically investigate various exchange-correlation functionals and various H concentrations, and we carefully track the various relevant H positions. For a dilute H concentration, the asymmetric oxygen site (Oa) is found to be energetically the most favorable. With increasing H content, the difference of the H adsorption energies between the terminal (Ot) and the Oa sites becomes smaller. Previous contradictions are ascribed mostly to the disregard of the H position along the Oa-Oa zig-zag chains in the intrabilayer region. Using the modern non-empirical strongly-constrained and appropriately-normed (SCAN) meta-generalized gradient approximation (GGA), the dilute-limit H adsorption energies are obtained as −2.89 eV/(H atom) and −2.97 eV/(H atom) in the bulk and on the surface, respectively, and the activation energy of H diffusion between the Oa sites as 0.11-0.15 eV/(H atom), consistent with previous experiments.