For an important special case, look up the topic of Hydrogen Embrittlement of Irons and Steels. This is not my field - I have a layman's interest for the reason discussed below, but I believe it illustrates some rough rules of thumb for when absorption into metals happens.
The main process whereby absorption happens begins when the $H_2$ bond is cloven by the metal - there is likely to be a pretty quantum mechanical description of this process whereby the atomic electrons interact with the conduction electron sea in the metal, but I shall leave another answer to give a reference as I am not aware of one and I am not a chemist, so this isn't my field. However, such cleaving is well known to happen experimentally, and the Wikipedia article on bond dissociation energy states that the $H_2$ bond is strong (indeed quite comparable to all the other bonds listed there) and only cloven by metals or strong oxidants. So this is the first requirement: most often the target crystal will be a metal or a strong oxidant.
The hydrogen atom is especially small so this helps the penetration of a proton electron pair into a lattice: the small size means that the charges are "well shielded" so the Coulombic potential barrier to penetration is going to be smallish. This characteristic also helps the second stage of the process whereby lone hydrogen atoms recombine in "voids" inside the metal. The $H_2$ molecule is small, thus helping this process too.
The Wikipedia article on Hydrogen Embrittlement also states the fairly obvious that the process is helped by high hydrogen pressures and temperatures, the former begetting high concentration gradients, the latter kinetic energy to raise the probability of tunnelling through Coulombic potential barriers.
This issue is an extremely important issue either in the design of the bottles and flasks for high pressure hydrogen, or in the design of "sponge" substances that can store hydrogen. At least one of these technologies is needed to make hydrogen fuel cell technology workable. One of the desirable characteristics of oil is that huge amounts of energy can be transported for little energy expenditure - an important property for mechanised agriculture. In contrast, some way to "package" energy is need for the use of e.g. solar energy for this purpose. The electrolysis of water by solar and storage of the hydrogen produced is one proposed solution to this problem.