Metal–organic frameworks (MOFs) have been widely studied for their ability to capture and store greenhouse gases. However, most computational discovery efforts study hypothetical MOFs without consideration of their stability, limiting the practical application of novel materials. We overcome this limitation by screening hypothetical ultrastable MOFs that have predicted high thermal and activation stability, as judged by machine learning (ML) models trained on experimental measures of stability. We enhance this set by computing the bulk modulus as a measure of mechanical stability and filter 1102 mechanically robust hypothetical MOFs from a database of ultrastable MOFs (USMOF DB). Grand Canonical Monte Carlo simulations are then employed to predict the gas adsorption properties of these hypothetical MOFs, alongside a database of experimental MOFs. We identify privileged building blocks that lead MOFs in USMOF DB to show exceptional working capacities compared to the experimental MOFs. We interpret these differences by training ML models on CO2 and CH4 adsorption in these databases, showing how poor model transferability between data sets indicates that novel design rules can be derived from USMOF DB that would not have been gathered through assessment of structurally characterized MOFs. We identify geometric features and node chemistry that will enable the rational design of MOFs with enhanced gas adsorption properties in synthetically realizable MOFs.