Despite the prospects of intrinsically porous planar nanomaterials in separation applications, their synthesis on a large scale remains challenging. In particular, preparing water-selective carbon nanomembranes (CNMs) from self-assembled monolayers (SAMs) is limited by the cost of epitaxial metal substrates and molecular precursors with specific chemical functionalities. In this work, we present a facile fabrication of CNMs from polycyclic aromatic hydrocarbons (PAHs) that are drop-cast onto arbitrary supports, including foils and metalized films. The electron-induced carbonization is shown to result in continuous membranes of variable thickness, and the material is characterized with a number of spectroscopic and microscopic techniques. Permeation measurements with freestanding membranes reveal a high degree of porosity, but the selectivity is found to strongly depend on the thickness. While the permeance of helium remains almost the same for 6.5 and 3.0 nm thick CNMs, water permeance increases by 2 orders of magnitude. We rationalize the membrane performance with the help of kinetic modeling and vapor adsorption experiments.