Hydrogen storage is expected to play a crucial role in the comprehensive defossilization of energy systems. In this context, the focus is typically on underground hydrogen storage (e.g., in salt caverns). However, aboveground storage, which is independent of geological conditions and might offer other technical advantages, could provide systemic benefits and, thereby, gain shares in the hydrogen storage market. Against this background, this paper examines the market relevance of aboveground compared to underground hydrogen storage. Using the open-source energy system model and optimization framework of Europe, PyPSA-Eur, the influence of geological independence, storage cost relations, and technical storage characteristics (i.e., efficiencies) on mentioned market relevance of aboveground hydrogen storage are investigated. Further, the expectable market relevance based on current cost projections for the future is assessed. The studies show that in terms of hydrogen capacities, aboveground hydrogen storage plays a considerably smaller role compared to underground hydrogen storage. Even when assuming comparatively low aboveground storage cost, it will not exceed 1.7% (1.9TWhH2,LHV) of total hydrogen storage capacities in a cost-optimal European energy system. Regarding the amounts of annually stored hydrogen, aboveground storage could play a larger role, reaching a maximum share of 32.5% (168TWhH2,LHVa-1) of total stored hydrogen throughout Europe. However, these shares are only achievable for low cost storage in particularly suited energy system supply configurations. For higher aboveground storage costs or lower efficiencies, shares drop below 10% sharply. The analysis identifies some especially influential factors for achieving higher market relevance. Besides storage costs, the demand-orientation of a particular aboveground storage system (e.g., hydrogen storage at demand pressure levels) plays an essential role in market relevance. Further, overall efficiency can be a beneficial factor. Still, current projections of future techno-economic characteristics show that aboveground hydrogen storage is too expensive or too inefficient compared to underground storage. Therefore, to achieve notable market relevance, rather drastic cost reductions beyond current expectations would be needed for all assessed aboveground hydrogen storage technologies.