Partial localization is the phenomenon of self-aggregation of mass into highdensity structures that are thin in one direction and extended in the others. We give a detailed study of an energy functional that arises in a simplified model for lipid bilayer membranes. We demonstrate that this functional, defined on a class of two-dimensional spatial mass densities, exhibits partial localization and displays the "solid-like" behaviour of cell membranes. Specifically, we show that density fields of moderate energy are partially localized, that is, resemble thin structures. Deviation from a specific uniform thickness, creation of "ends," and the bending of such structures all carry an energy penalty, of different orders in terms of the thickness of the structure. These findings are made precise in a Gamma-convergence result. We prove that a rescaled version of the energy functional converges in the zero-thickness limit to a functional that is defined on a class of planar curves. Finiteness of the limit enforces both optimal thickness and non-fracture; if these conditions are met, then the limit value is given by the classical elastica (bending) energy of the curve.