We report gas and water flow measurements through microfabricated membranes with sub-2 nm aligned carbon nanotubes as pores. The measured gas flow exceeds predictions of the Knudsen diffusion model by more than an order of magnitude. The measured water flow exceeds values calculated from continuum hydrodynamics models by more than three orders of magnitude and is comparable to flow rates extrapolated from molecular dynamics simulations. The gas and water permeabilities of these nanotube-based membranes are several orders of magnitude higher than those of commercial polycarbonate membranes, despite having order of magnitude smaller pore sizes. These properties should enable fundamental studies of mass transport in confined environments, as well as more energy-efficient nanoscale filtration.