Yes, some gases can diffuse into and through metal. It is the bane of the high-vacuum engineer's life. Hydrogen is the worst because it tends to dissociate into atoms at the surface and the nucleus, a single proton, can then leave its electron behind and wander through the metal lattice until it picks up another electron when it leaves.
For example Mu-metal, favoured for some applications, typically has to be annealed in hydrogen at high temperature. Once that is over, it can take weeks or months for the residual hydrogen to diffuse out of the metal before a high enough vacuum can be achieved and the work proceed.
A "virtual leak" occurs where a small bubble of gas is embedded in the material inside a vacuum chamber. The leak usually happens because a tiny hole exists for the gas to diffuse out through, but sometimes the "hole" is no more than an ultra-thin skin of metal (invisible to the frustrated technician) and the gas diffuses through it. These little horrors can keep going for months or even years and generally mean replacing suspected parts and pumping down over and over again until the dodgy one is finally stumbled on.
Helium is both monatomic and the physically smallest atom. It can diffuse more easily than any other neutral atom or molecule, making certain metal foils unsuitable as say gas-tight liners for airships. As noted in another answer, in quantity it can also affect the bulk properties of the metal.
On a more energetic scale, hydrogen and helium nuclei (protons and alpha particles) can pass through thin metal foils if fired with sufficient energy, and this has been used to establish the crystalline structures of some metals and alloys (where, for whatever reason, electrons were unsuitable).
Other gases have much larger atoms (neon and other noble gases) or molecules (nitrogen and other diatomic molecules, water and other hydrides), but they can still diffuse extremely slowly through some metals. This can limit the lifetime of some microchips. A related phenomenon occurs where there is a defect in the lattice at the surface, such as a grain boundary, and a gas atom attaches to it. Defects are sometimes quite mobile and can migrate through the lattice; the gas atom will stabilise the defect and may be able to hitch a ride.
Quantum processes such as tunnelling are not really relevant, as they work over distances smaller than the atomic wavelength, which in turn is typically far smaller than the thickness of any metal atom or foil. The probability of a gas atom tunnelling across is so infinitesimal as to be effectively zero.