Graphene and other two-dimensional materials offer a new class of ultrathin membranes that can have atomically defined nanopores with diameters approaching those of hydrated ions. These nanopores have the smallest possible pore volumes of any ion channel, which, due to ionic dehydration and electrokinetic effects, places them in a novel transport regime and allows membranes to be created that combine selective ionic transport with ultimate permeance and could lead to separations and sensing applications. However, experimental characterization and understanding of sub-continuum ionic transport in nanopores below 2 nm is limited. Here we show that isolated sub-2 nm pores in graphene exhibit, in contrast to larger pores, diverse transport behaviours consistent with ion transport over a free-energy barrier arising from ion dehydration and electrostatic interactions. Current-voltage measurements reveal that the conductance of graphene nanopores spans three orders of magnitude and that they display distinct linear, voltage-activated or rectified current-voltage characteristics and different cation-selectivity profiles. In rare cases, rapid, voltage-dependent stochastic switching is observed, consistent with the presence of a dissociable group in the pore vicinity. A modified Nernst-Planck model incorporating ion hydration and electrostatic effects quantitatively matches the observed behaviours.