In the previous studies by the authors and others, it was demonstrated that there are two possible defect growth modes and a characteristic material length for any soft material. For a pre-existing defect smaller than the material characteristic length, the energy is dissipated all around the defect as it grows and the critical load for the growth is independent of the defect size. For defects larger than the characteristic length, the growth is by cracking and the energy is dissipated along a plane. Thus, the critical load for the growth is size dependent and can be predicted by fracture mechanics. In this study, we apply the same energy-based argument to the failure of thin membranes, with the focus on the first growth mode that gives the maximum critical load. We assume that strain localization due to damage is the precursor to rupture, and hence, we model the corresponding zone as a through-thickness hole, with its size smaller than the material characteristic length. The defect grows when the elastic energy relaxed by the growth is enough to provide the energy needed for internal microstructure changes. This leads us to the size-independent failure conditions for membranes under the biaxial load. The conditions are expressed in terms of either two principal stretches or two principal stresses for two different types of materials. For verification, we test the theory using the published experimental data on natural and styrene-butadiene rubber. By using the experimental data from equal biaxial loading, we predict the critical principal stretch ratios and critical stresses for different biaxialities. The predictions agree well with the experimental results.