Novel carbon-based periodic networks of crossed nanotubes with fullerene junctions (FNT-nets) has been designed as a potential solid hydrogen storage medium. Density functional theory calculations and molecular dynamics simulations have been performed to explore the structural stability and the hydrogen-storage potential of FNT-nets. Via the B-doping strategy, the binding energies of the alkali metal (Li, Na) and alkaline-earth metal (Be) were effectively improved to 2.47, 2.10, and 1.98 eV, respectively. The Li, Na, and Be atoms could be firmly dispersed atomically on the surface of B-FNT-nets, donating 0.732, 0.711, and 1.455 e to the substrate. The positive charge on the metal proved to be proportional to the strength of H-2 polarization. Overall, up to four hydrogen molecules could be stored around one lithium with an average adsorption energy of 0.20 eV. Five hydrogen molecules were found to be around sodium and beryllium atoms with 0.17 and 0.27 eV in the range of reversible physical adsorption. The corresponding theoretical uptakes for Li@C158B22, Na@C132B24, and Be@C132B24 systems (13.95, 10.09, and 10.85 wt%, respectively) exceed the ultimate goal (6.5 wt%) as laid out by US Department of Energy. Our results reveal that the FNT-nets are good candidates for hydrogen storage.