Mass transfer across porous materials with nanoscale thickness is of great interest in terms of both fundamentals of fluid dynamics and practical challenges of membrane separation. In particular, few-atom thick sieves are viewed as attractive candidates to achieve ultimate permeability without compromising membrane selectivity. In this work, we introduce a vacuum system for studying vapour and gas permeation in micrometre-sized samples of suspended nanometre-thick films. Steady-state permeation rates are measured with a mass-spectrometer directly connected to the downstream side of a membrane cell. A built-in nanoaperture is used as a reference to calibrate the detector in situ. A feed compartment is designed in a way that allows for preparing gaseous mixtures of variable composition, including vapours of volatile liquids. Room-temperature measurements with carbon nanomembranes confirm that this material is selective to water vapour and can efficiently separate it from mixtures with a variety of gases and organic compounds. We demonstrate that a high permeance for water is maintained regardless of the molar fraction and discuss its strong pressure dependence by invoking adsorption-related formalism.