In this study, we have employed density functional theory with a range of van der Waals corrections to study geometries, electronic structures, and hydrogen (H2) storage properties of carbon ene-yne (CEY) decorated with selected alkali (Na, K) and alkaline-earth metals (Mg, Ca). We found that all metals, except Mg, bind strongly by donating a major portion of their valence electrons to the CEY monolayers. Thermal stabilities of representative systems, Ca-decorated CEY monolayers, have been confirmed through ab initio molecular dynamics simulations (AIMD). We showed that each metal cation adsorbs multiple H2 with binding energies (Ebind) considerably stronger than on pristine CEY. Among various metal dopants, Ca stands out with the adsorption of five H2 per each Ca having Ebind values within the desirable range for effective adsorption/desorption process. The resulting gravimetric density for CEY@Ca has been found around 6.0 wt % (DFT-D3) and 8.0 wt % (LDA), surpassing the U.S. Department of Energy's 2025 goal of 5.5 wt %. The estimated H2 desorption temperature in CEY@Ca exceeds substantially the boiling point of liquid nitrogen, which confirms its potential as a practical H2 storage medium. We have also employed thermodynamic analysis to explore the H2 adsorption/desorption mechanism at varied conditions of temperature and pressure for real-world applications.