Carbon-based platforms for hydrogen storage are attractive due to the stability of carbon allotropes, as well as the energetically efficient physisorption mechanisms of hydrogen to carbon surfaces. Hydrogen adsorption on fullerenes, graphene, and carbon nanotubes have been well studied, and it is known that the hydrogen storage is limited by the accessible surface area. Here, we propose a novel fullerene-like molecule—a so-called fulleryne—to increase potential hydrogen storage capacity of carbon-based systems. Fullerynes are spherical molecules characterized by acetylenic substitution in the aromatic bond structure of fullerenes. The result is a less dense, more porous structure. Here, via full atomistic molecular dynamics (MD) simulation, we characterize the energetic stability and properties of fullerynes (single acetylenic link) and fullerdiynes (a double acetylenic link), including self-adhesion and bulk modulus, and compare to fullerenes. We then quantify hydrogen (H₂) adsorption energy, and assess the storage capacity (via accessible surface area) and mobility (via hydrogen diffusivity). We find that the sparse, lightweight fullerdiyne systems has relatively high specific hydrogen accessible surface area, near equivalent adsorption energy as graphene/fullerene, and facilitates hydrogen diffusion by enabling motion through the interior of the spherical structure.