“if you had cheap batteries that could store electricity for longer than a few hours and maintain their charge.”
The battery in your phone can keep a charge for months.
Absolutely. It stores enough for an extended amount of time, and the amount needed to generate the electricity it uses is minimal, so they can miniaturize it for the phone. The model can be used for anything, however if space requirements can be miniaturized, the model is the same. In reality, one has to increase the amount of generation, or increase the size and time needed for storage. Considering the advance in nuclear reactors and there size, I can see eventually mini nukes to operate an entire household. But what would be the profit in that?