It sounds like quite an order, but this sort of functionality should be ubiquitous. Calculators must employ very similar circuitry. I would go tear one apart to analyze how it's done, except they've become too miniaturized, complex, and often potted.
I picture using a rechargeable single cell lithium ion battery. Likely in a 18650 or 26650 package.
There are indeed prebuilt circuits that perform individual tasks, such as this convenient USB charger:
**broken link removed**
and this solar harvester:
**broken link removed**
The trick will be, how to tie everything together to make it behave like I described.