First-principles density functional and quantum Monte Carlo calculations of light-element doped fullerenes reveal significantly enhanced molecular H-2 binding for substitutional B and Be. A nonclassical three-center binding mechanism between the dopant and H-2 is identified, which is maximized when the empty p(z) orbital of the dopant is highly localized. The calculated binding energies of 0.2-0.6 eV/H-2 is suited for reversible hydrogen storage at near standard conditions. The calculated H-2 sorption process is barrierless, which could also significantly simplify the kinetics for the storage.