Could food storage prevent a depression? In Genesis chapter 41 Joseph interpreted the dreams of Pharaoh about 7 years of plenty and 7 years of famine which resulted in saving Egypt from famine, among other things.

I think this is a great economic feat to say the least. But it brings up so many economic questions about the scenario and how it might play out under different economic models and policy decisions.

First off, nowadays can a big enough food storage keep a city or a nation from going into an economic depression? Which economic models might I use to analyze the situation? The Great Depression included the Dust Bowl, and a lot of hunger, and it makes me wonder what difference a gigantic food storage could have done. 

Egypt in that Bible story was experiencing a famine though, and economic depressions don't seem to be caused by famines necessarily. So is food storage only applicable during a famine?
Keep in mind that during the 7 years of famine in the Bible, there was still a negative production shock that more than likely produced a heavy recession for some of those seven years as workers tried to reallocate their labor to something more useful than agriculture. A recession merely represents a fall in final goods production. The grain that was saved up doesn't represent production for that period (rather it would be for the previous seven years of economic boom). The grain stored up represents savings rather than production.

It doesn't look like Joseph or Pharaoh told the people that the famine was coming (imagine the panic), so workers wouldn't have predicted the change in permanent income, and their intertemporal substitution would be much smaller because of the error in their adaptive/rational expectations. Joseph merely used taxes (one-fifth of each year's harvest) to engage in consumption smoothing.

So to answer most of your questions, food storage doesn't prevent negative output shocks. It merely is a hedge against them. Arguably most people should try to have emergency savings of their own to hedge against recessions of their own, but especially in America, we are not very good at saving money. We also have a lot of population who live paycheck to paycheck, and are liquidity constrained, unable to consumption smooth according to their predictions in permanent income, even if they want to.