To achieve the sustainability goals set for the European aviation sector, hydrogen-powered solutions are currently being investigated. Storage solutions are of particular interest, with liquid hydrogen tanks posing numerous challenges with regard to the structural integrity of materials at cryogenic temperatures, as well as safety issues because of the high flammability of hydrogen. In this context and in the scope of the Horizon 2020 Clean Aviation Joint Undertaking (CAJU) project H2ELIOS, the gas permeability behavior of prepreg tape carbon fibre reinforced polymer (CFRP) material was studied. Investigations were performed after thermal shock to 20 K (liquid hydrogen immersion) as well as after a uniaxial stress application at 77 K to identify the shift from Fickian behavior after diverse aging conditions. Helium gas permeation was tested at room temperature (RT), and its representativeness to hydrogen permeation in a range of temperatures was considered in the study. The material’s permeation behavior was compared to ideal Fickian diffusion as a means of identifying related permeation barrier function degradation. Finally, it was possible to identify Fickian, near-Fickian, and non-Fickian behaviors and correlate them with the material’s preconditioning.