Recently a Nahm transform has been discovered for magnetic bags, which are conjectured to arise in the large n limit of magnetic monopoles with charge n. We interpret these ideas using string theory and present some partial proofs of this conjecture. We then extend the notion of bags and their Nahm transform to higher gauge theories and arbitrary domains. Bags in four dimensions conjecturally describe the large n limit of n self-dual strings. We show that the corresponding Basu-Harvey equation is the large n limit of an equation describing n M2-branes, and that it has a natural interpretation in loop space. We also formulate our Nahm equations using strong homotopy Lie algebras.