By the Freyd-Mitchell Embedding Theorem, any Abelian category admits an exact embedding into the category of modules over some ring. I'm not (currently) hoping to learn a proof, but instead I want to know if there are specific cases of this, aside from the obvious ones, which we can work out explicitly.
For example: is there a reasonably explicit way to describe the category of sheaves of Abelian groups on some space as a module category? Or the category of chain complexes of Abelian groups? Let's say the threshold for explicit is, at minimum, that we can cleanly describe the ring over which everything is a module.