We report rigorous quantum five-dimensional (5D) calculations of the coupled translation-rotation (T-R) eigenstates of a H(2) molecule adsorbed in metal organic framework-5 (MOF-5), a prototypical nanoporous material, which was treated as rigid. The anisotropic interactions between H(2) and MOF-5 were represented by the analytical 5D intermolecular potential energy surface (PES) used previously in the simulations of the thermodynamics of hydrogen sorption in this system [Belof et al., J. Phys. Chem. C 113, 9316 (2009)]. The global and local minima on this 5D PES correspond to all of the known binding sites of H(2) in MOF-5, three of which, α-, β-, and γ-sites are located on the inorganic cluster node of the framework, while two of them, the δ- and ε-sites, are on the phenylene link. In addition, 2D rotational PESs were calculated ab initio for each of these binding sites, keeping the center of mass of H(2) fixed at the respective equilibrium geometries; purely rotational energy levels of H(2) on these 2D PESs were computed by means of quantum 2D calculations. On the 5D PES, the three adjacent γ-sites lie just 1.1 meV above the minimum-energy α-site, and are separated from it by a very low barrier. These features allow extensive wave function delocalization of even the lowest translationally excited T-R eigenstates over the α- and γ-sites, presenting significant challenges for both the quantum bound-state calculations and the analysis of the results. Detailed comparison is made with the available experimental data.