I'm not really concerned about React-specific implementation, but rather a data structure that serves as a facade over the storage layer (either a remote server or the storage systems made available to the browser). This way the UI (e.g. React components and other view-like things) just talks to this facade and doesn't have to know anything about the network, its availability, or browser-specific storage options.
Mozilla's localForage is a good tool specifically for abstracting away the client-side storage mechanism from the UI logic. Aside from that I have looked at Flux and its stores would be the place where you would implement such a facade.