Hard carbons are the leading candidate anode materials for sodium-ion batteries. However, the sodium-insertion mechanisms remain under debate. Here, employing a novel analysis of operando and ex situ pair distribution function (PDF) analysis of total scattering data, supplemented by information on the local electronic structure provided by operando 23Na solid-state NMR, we identify the local atomic environments of sodium stored within hard carbon and provide a revised mechanism for sodium storage. The local structure of carbons is well-described by bilayers of curved graphene fragments, with fragment size increasing, and curvature decreasing with increasing pyrolysis temperature. A correlation is observed between the higher-voltage (slope) capacity and the defect concentration inferred from the size and curvature of the fragments. Meanwhile, a larger lower-voltage (plateau) capacity is observed in samples modeled by larger fragment sizes. Operando PDF data on two commercially relevant hard carbons reveal changes at higher-voltages consistent with sodium ions stored close to defective areas of the carbon, with electrons localized in the antibonding π*-orbitals of the carbon. Metallic sodium clusters approximately 13-15 Å in diameter are formed in both carbons at lower voltages, implying that, for these carbons, the lower-voltage capacity is determined by the number of regions suitable for sodium cluster formation, rather than by having microstructures that allow larger clusters to form. Our results reveal that local atomic structure has a definitive role in determining storage capacity, and therefore the effect of synthetic conditions on both the local atomic structure and the microstructure should be considered when engineering hard carbons.