Soft robots are known for their ability to comply and having superior extensibility. However, one of the limitations of most of these robots is that they can stand only a limited amount of load before buckling, and they feature a non-negligible initial height. Hybrid soft-rigid actuators seem to offer a trade-off between compliance and the amount of load they can withstand, but only a few simple models have been proposed to describe the behavior of these actuators. In this paper, we propose a design, model and experimental validation of a Hyperelastic Ballooning Membrane actuator (HBMA) which shows an extensibility higher than 179%, as well as an ability to stand more than 20 times its own weight. Two models, giving the dynamic behavior of the HBMA in terms of displacement and pressure, have been derived from different hyperelastic models (Neo-Hookean and Mooney-Rivlin) and compared in terms of accuracy and robustness. Finally, an example of a hybrid soft-rigid continuum ballooning robot built with HBMAs is presented and characterized experimentally.