Both LiBH4 and NaBH4 are well known for having high hydrogen contents, but also high decomposition temperatures and slow hydrogen absorption–desorption kinetics, preventing their use for hydrogen storage applications. The low melting temperature (219 °C) of their eutectic mixture 0.71 LiBH4–0.29 NaBH4 allowed the synthesis of a new composite material through the melt infiltration of the hydrides into the ~5 nm diameter pores of a CMK-3 type carbon. A composite of 0.71 LiBH4–0.29 NaBH4 and non-porous graphitic carbon discs was also prepared by similar methods for comparison. Both composites showed improved kinetics and a partial reversibility of the dehydrogenation/rehydrogenation reactions. However, the best results were observed for the CMK-3 nanoconfined hydrides; a consistent uptake of about 3.5 wt.% H2 was recorded after five hydrogenation/dehydrogenation cycles for an otherwise non-reversible system. The improved hydrogen release kinetics are attributed to carbon–hydride surface interactions rather than nanoconfinement, while enhanced heat transfer due to the carbon support may also play a role. Likewise, the carbon–hydride contact proved beneficial in terms of reversibility, without, however, ruling out the potential positive effect of pore confinement.