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We determine the energy storage needed to achieve self sufficiency to a given reliability as a function of excess capacity in a combined solar-energy generation and storage system. Based on 40 years of solar-energy data for the St. Louis region, we formulate a statistical model that we use to generate synthetic insolation data over millions of years. We use these data to monitor the energy depletion in the storage system near the winter solstice. From this information, we develop explicit formulas for the required storage and the nature of cost-optimized system configurations as a function of reliability and the excess generation capacity. Minimizing the cost of the combined generation and storage system gives the optimal mix of these two constituents. For an annual failure rate of less than 3%, it is sufficient to have a solar generation capacity that slightly exceeds the daily electrical load at the winter solstice, together with a few days of storage.