Hydromedusan jellyfish swim by rhythmic pulsation of their mesogleal bells. A single swimming muscle contracts to create thrust by ejecting water from the subumbrellar cavity. At the end of the contraction, energy stored in the deformation of the mesogleal bell powers the refilling stage, during which water is sucked back into the subumbrellar cavity. The mesoglea is a mucopolysaccharide gel reinforced with radially oriented fibres made primarily of a protein homologous to mammalian fibrillin. Most of the energy required to power the refill stroke is thought to be stored by stretching these fibres. The elastic modulus of similar fibrillin-rich fibres has been measured in other systems and found to be in the range of 0.2 to 1.1 MPa. In this paper, we measured the diameters of the fibres, their density throughout the bell, and the mechanical behaviour of the mesoglea, both in isolated samples and in an intact bell preparation. Using this information, we calculated the stiffness of the fibres of the hydromedusa Polyorchis penicillatus, which we found to be approximately 0.9 MPa, similar in magnitude to other species. This value is two orders of magnitude more compliant than the stiffness of the component fibrillin microfibrils previously reported. We show that the structure of the radial fibres can be modelled as a parallel fibre-reinforced composite and reconcile the stiffness difference by reinterpreting the previously reported data. We separate the contributions to the bell elasticity of the fibres and mesogleal matrix and calculate the energy storage capacity of the fibres using the calculated value of their stiffness and measured densities and diameters. We conclude that there is enough energy potential in the fibres alone to account for the energy required to refill the subumbrellar cavity.