We show that gel-phase lipid membranes soften upon bending, leading to curvature localization and a negative compressibility. Using simulations of two very different lipid models to quantify shape and stress strain relation of buckled membranes, we demonstrate that gel phase bilayers do not behave like Euler elastica and hence are not well described by a quadratic Helfrich Hamiltonian, much unlike their fluid-phase counterparts. We propose a theoretical framework which accounts for the observed softening through an energy density that smoothly crosses over from a quadratic to a linear curvature dependence beyond a critical new scale l(-1). This model captures both the shape and the stress strain relation for our two sets of simulations and permits the extraction of bending moduli, which are found to be about an order of magnitude larger than the corresponding fluid phase values. We also find surprisingly large crossover lengths 1, several times bigger than the bilayer thickness, rendering the exotic elasticity of gel-phase membranes more strongly pronounced than that of homogeneous compressible sheets and artificial metamaterials. We suggest that such membranes have unexpected potential as nanoscale systems with striking materials characteristics.