Membranes with 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 and redox flow batteries. Here we report a new approach to designing solution-processable ion-selective polymer membranes with both intrinsic microporosity and ion-conductive functionality. This was achieved by synthesizing polymers 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 facilitated transport of protons and cations through these membranes, as well as 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. This membrane design strategy paves the way for producing a new-generation of ion-exchange membranes for electrochemical energy conversion and storage applications.