Membranes are widely used for separation processes in applications such as water desalination, batteries and dialysis, and are crucial in key sectors of our economy and society1. The majority of technologically exploited membranes are based on solid polymers and function as passive barriers, whose transport characteristics are governed by their chemical composition and nanostructure. Although such membranes are ubiquitous, it has proved challenging to maximize selectivity and permeability independently, leading to trade-offs between these pertinent characteristics2. Self-assembled biological membranes, in which barrier and transport functions are decoupled3,4, provide the inspiration to address this problem5,6. Here we introduce a self-assembly strategy that uses the interface of an aqueous two-phase system to template and stabilize molecularly thin (approximately 35 nm) biomimetic block copolymer bilayers of scalable area that can exceed 10 cm2 without defects. These membranes are self-healing, and their barrier function against the passage of ions (specific resistance of approximately 1 MΩ cm2) approaches that of phospholipid membranes. The fluidity of these membranes enables straightforward functionalization with molecular carriers that shuttle potassium ions down a concentration gradient with exquisite selectivity over sodium ions. This ion selectivity enables the generation of electric power from equimolar solutions of NaCl and KCl in devices that mimic the electric organ of electric rays.