There are materials that absorb methane under ambient conditions. A quick Google Scholar search for "metal organic framework methane" shows some of the most recent and popular synthesized materials that can do this. Dr. Shengqian Ma at University of South Florida is a big figure in the field, and he currently collaborates with my computational research group. This is a figure of methane sorption for a material his group made at different temperatures:
Granted the highest uptakes are at high pressures (1 bar ~= 1 atm), and the graph is somewhat muddled at ~1 atm, but this material sorbs (much) more than its own volume at that pressure, and there are other examples.
It is a somewhat new class of materials (~20 years). One of the first to be synthesized is MOF-5, which wasn't that useful but served as a good prototype example for future syntheses. But there are a ton of them that have been synthesized to date. They are crystalline materials that are made by self-assembly of metal cations and organic ligands (usually). Some have even been made with light elements e.g. beryllium.
All of my graduate research thus far (1.5 years) has been studying the mechanism of gas sorption in MOFs. Many can physisorb methane, hydrogen, carbon dioxide, nitrogen, and other gases quantitatively. To be specific, several can do so with methane at ambient T/P. Many copper MOFs, for example, induce a pretty strong interaction (thus binding effect) with hydrogen gas, which is not easy to contrive.
See the wiki also.wiki