We present the results of a density functional theory study of sodium storage and mobility on graphdiyne (GDY) and consider the applicability of GDY intercalated with sodium (Na) as an anode material for rechargeable batteries. The maximum capacity, energy barriers for Na diffusion throughout the layers, and expansion of the layers due to Na insertion are determined. The calculations indicate that Na intercalates within the GDY bulk layers with a capacity of NaC5.14 without expansion (316 mA h g−1) and NaC2.57 with expansion of 28% (497 mA h g−1). The energy barrier for movement of Na in the slit pore formed by two GDY bulk layers is found to be 0.82 eV for bulk GDY with an AB-2 stacking, and the barrier for movement through a GDY sheet is found to be 0.12 eV. The barrier for movement in the slit pore formed by sheets becomes even lower for AB-3 stacking, with values of 0.68 and 0.40 eV found for different pathways. Movement from one GDY sheet to another for the AB-3 stacking also has a moderate energy of 0.37 eV. Therefore, GDY intercalated with Na is proposed to have potential as an anode material for rechargeable batteries.