Solid state materials capable of storing hydrogen with high gravimetric (9 wt %) and volumetric density (70 g/L) are critical for the success of a new hydrogen economy. In addition, an ideal storage system should be able to operate under ambient thermodynamic conditions and exhibit fast hydrogen sorption kinetics. No materials are known that meet all these requirements. While recent theoretical efforts showed some promise for transition-metal-coated carbon fullerenes, later studies demonstrated that these metal atoms prefer to cluster on the fullerene surface, thus reducing greatly the weight percentage of stored hydrogen. Using density functional theory we show that Li-coated fullerenes do not suffer from this constraint. In particular, we find that an isolated Li12C60 cluster where Li atoms are capped onto the pentagonal faces of the fullerene not only is very stable but also can store up to 120 hydrogen atoms in molecular form with a binding energy of 0.075 eV/H-2. In addition, the structural integrity of Li12C60 clusters is maintained when they are allowed to interact with each other. The lowest energy structure of the dimer is one where the Li atom capped on the five-member ring of one fullerene binds to the six-member ring of the other. The binding of hydrogen to the linking Li atom and the potential of materials composed of Li12C60 building blocks for hydrogen storage are discussed.