Hydra vulgaris has long been known for its remarkable regenerative capabilities. As such, they are an historical inspiration for reaction-diffusion systems trying to explain their spontaneous patterning. Recently, it became clear that their patterning is an integrated mechano-chemical process where morphogen dynamics is influenced by tissue mechanics. One roadblock to understand their self-organization is our lack of knowledge about the mechanical properties of these organisms. In this paper, we combined microfluidic developments to perform parallelized microaspiration rheological experiments and numerical simulations to fully characterize these mechanical properties. We found three different behaviors depending on the amount of applied stresses: an elastic response, a visco-elastic one and tissue rupture. Using rheological models of deformable shells, we quantify their elastic modulus, effective viscosity as well as the critical stresses required to switch between behaviors. Based on these experimental results, we propose a full description of the internal tissue mechanics during normal regeneration. Our results provide the first step towards the development of original mechano-chemical models of patterning grounded in quantitative, experimental data.