Hydrogen has been widely considered to hold promise for solving challenges associated with the increasing demand for green energy. While many chemical and biochemical processes produce molecular hydrogen as byproducts, electrochemical approaches using water electrolysis are considered to be a predominant method for clean and green hydrogen production. We discuss the current state-of-the-art in molecular hydrogen production and storage and, more significantly, the increasing role of computational modeling in predictively designing and deriving insights for enhancing hydrogen storage efficiency in current and future materials of interest. One of the key takeaways of this review lies in the continued development and implementation of large-scale atomistic simulations to enable the use of designer electrolyzer–electrocatalysts operating under targeted thermophysical conditions for increasing green hydrogen production and improving hydrogen storage in advanced materials, with limited tradeoffs for storage efficiency.