Oxide membranes are the foundation of several electrochemical devices and sensors, where functionality is related to selective transport of electrons and ions through a membrane or physical responses from an external perturbation. The ability to engineer power sources and sensors for the rapidly growing field of autonomous systems requires high power density and specific energy. Clamped free-standing nanoscale membranes provide an experimentally tunable platform to explore the limits of dimensionality reduction for such purposes. This review addresses the following: (i) advancing experimental methods to fabricate nanoscale oxide membranes that can sustain a chemical potential gradient, thermomechanically stable under large thermal cycles, and can be electrically interrogated with negligible parasitic loss; (ii) a representative example of high performance energy devices, solid oxide fuel cells, utilizing such membranes; and (iii) a brief discussion on emerging research directions broadly in the areas of condensed matter sciences and energy conversion and storage intersecting low-dimensional complex oxide materials.