The permeability barrier of nuclear pore complexes (NPCs) controls the exchange between nucleus and cytoplasm. It suppresses the flux of inert macromolecules > or = 30 kDa but allows rapid passage of even very large cargoes, provided these are bound to appropriate nuclear transport receptors. We show here that a saturated hydrogel formed by a single nucleoporin FG-repeat domain is sufficient to reproduce the permeability properties of NPCs. Importin beta and related nuclear transport receptors entered such hydrogel >1000x faster than a similarly sized inert macromolecule. The FG-hydrogel even reproduced import signal-dependent and importin-mediated cargo influx, allowing importin beta to accelerate the gel entry of a large cognate cargo more than 20,000-fold. Intragel diffusion of the importin beta-cargo complex occurred rapidly enough to traverse an NPC within approximately 12 ms. We extend the "selective phase model" to explain these effects.