Yes indeed the energy of a gamma ray can be stored by a nucleus. It could, for example, set the nucleus spinning, creating what's called a nuclear isomer. Isomers are nuclei with the same number of both protons and neutrons, but with different energy levels due to one being excited above the ground state by, for example, having more angular momentum (it's spinning). Such isomers are usually very unstable (with a short-half life) and will quickly revert to the ground state, emitting a gamma ray or two in the process.

Some isomers, however, have a very long half-life, storing large amounts of energy for long periods. Proposed application are bombs and gamma ray lasers. Either one would be hard to do.