The behaviour of electron gases in restricted geometries provides a means to explore the fundamental quantum-mechanical properties of fermion gases at mesoscopic length scales . But the existence of Coulomb repulsion between electrons unavoidably complicates the physics. Quantum gases of neutral fermions-such as3 He quasiparticles in a dilute solution of3 He in4 He, cooled to millikelvin temperatures -therefore offer a means of probing regimes completely inaccessible to electronic systems. Here we demonstrate the quantum exclusion of a3 He fermion gas from a network of narrow channels, connected to a reservoir of (3) He/(4) He solution. The effect is expected from simple quantum-mechanical arguments, which predict that the3 He atoms cannot enter the channels when their wavelength exceeds [radical] 2 times the channel width. By adjusting the temperature of the solution, the energy of the particles and hence their average wavelength can be controlled. In this way, we observe temperature-dependent changes in the penetration of the3 He quasiparticles into the channels. Our results demonstrate the macroscopic response of an atomic gas to basic quantum-mechanical restrictions at the mesoscopic level.