Using hydrogen as an energy carrier requires new technological solutions for its onboard storage. The exploration of two-dimensional (2D) materials for hydrogen storage technologies has been motivated by their open structures, which facilitates fast hydrogen kinetics. Herein, the hydrogen storage properties of lightweight metal functionalized r57 haeckelite sheets are studied using density functional theory (DFT) calculations. H2 molecules are adsorbed on pristine r57 via physisorption. The hydrogen storage capacity of r57 is improved by decorating it with alkali and alkaline-earth metals. In addition, the in-plane substitution of r57 carbons with boron atoms (B@r57) both prevents the clustering of metals on the surface of 2D material and increases the hydrogen storage capacity by improving the adsorption thermodynamics of hydrogen molecules. Among the studied compounds, B@r57-Li4, with its 10.0 wt% H2 content and 0.16 eV/H2 hydrogen binding energy, is a promising candidate for hydrogen storage applications. A further investigation, as based on the calculated electron localization functions, atomic charges, and electronic density of states, confirm the electrostatic nature of interactions between the H2 molecules and the protruding metal atoms on 2D haeckelite sheets. All in all, this work contributes to a better understanding of pure carbon and B-doped haeckelites for hydrogen storage.