By means of dissipative particle dynamics (DPD) and Monte Carlo (MC) simulations, we explored geometrical, transport, and sorption properties of hydrated Nafion-type polyelectrolyte membranes. Composed of a perfluorinated backbone with sulfonate side chains, Nafion self-assembles upon hydration and segregates into interpenetrating hydrophilic and hydrophobic subphases. This segregated morphology determines the transport properties of Nafion membranes that are widely used as compartment separators in fuel cells and other electrochemical devices, as well as permselective diffusion barriers in protective fabrics. We introduced a coarse-grained model of Nafion, which accounts explicitly for polymer rigidity and electrostatic interactions between anionic side chains and hydrated metal cations. In a series of DPD simulations with increasing content of water, a classical percolation transition from a system of isolated water clusters to a 3D network of hydrophilic channels was observed. The hydrophilic subphase connectivity and water diffusion were studied by constructing digitized replicas of self-assembled morphologies and performing random walk simulations. A non-monotonic dependence of the tracer diffusivity on the water content was found. This unexpected behavior was explained by the formation of large and mostly isolated water domains detected at high water content and high equivalent polymer weight. Using MC simulations, we calculated the chemical potential of water in the hydrated polymer and constructed the water sorption isotherms, which extended to the oversaturated conditions. We determined that the maximum diffusivity and the onset of formation of large water domains corresponded to the saturation conditions at 100% humidity. The oversaturated membrane morphologies generated in the canonical ensemble DPD simulations correspond to the metastable and unstable states of Nafion membrane that are not realized in the experiments.