Fundamental biological processes are carried out by curved epithelial sheets that enclose a pressurized lumen. How these sheets develop and withstand three-dimensional deformations has remained unclear. Here we combine measurements of epithelial tension and shape with theoretical modelling to show that epithelial sheets are active superelastic materials. We produce arrays of epithelial domes with controlled geometry. Quantification of luminal pressure and epithelial tension reveals a tensional plateau over several-fold areal strains. These extreme strains in the tissue are accommodated by highly heterogeneous strains at a cellular level, in seeming contradiction to the measured tensional uniformity. This phenomenon is reminiscent of superelasticity, a behaviour that is generally attributed to microscopic material instabilities in metal alloys. We show that in epithelial cells this instability is triggered by a stretch-induced dilution of the actin cortex, and is rescued by the intermediate filament network. Our study reveals a type of mechanical behaviour-which we term active superelasticity -that enables epithelial sheets to sustain extreme stretching under constant tension.