Membranes which allow fast and selective transport of protons and cations are required for a wide range of electrochemical energy conversion and storage devices, such as proton-exchange membrane (PEM) fuel cells (PEMFCs) and redox flow batteries (RFBs). Herein we report a new approach to designing solution-processable ion-selective polymer membranes with both intrinsic microporosity and ion-conductive functionality. Polymers are synthesized with rigid and contorted backbones, which incorporate hydrophobic fluorinated and hydrophilic sulfonic acid functional groups, to produce membranes with negatively charged subnanometer-sized confined ionic channels. The ready transport of protons and cations through these membranes, and the high selectivity towards nanometer-sized redox-active molecules, enable efficient and stable operation of an aqueous alkaline quinone redox flow battery and a hydrogen PEM fuel cell.