If stored properly, root veg and certain other produce will last for at least a few months, though they will certainly deteriorate in quality over that time. All they would need was enough to keep them over the winter, until they had fresh seasonal stuff growing again. Most folks before year-round produce sections had to sort through their stored veg over the winter and get rid of all the spoiling stuff so it wouldn't get the rest of it rotting. By spring, pickings would have been very slim, and probably pretty flabby and blah, but still edible.
But yes, there were a lot of agricultural inaccuracies in the show which I take a strangely self-righteous pleasure in noting. First of all, they wouldn't be harvesting every single veg at the same time -- green beans and potatoes, for instance, have different seasons. Beans at that late date would more have been mature, and then you harvest the whole vine and hang them to dry to shell them later, you don't pick them individually and store them in boxes.
And after a storm like that, the most important thing would be to get up the aboveground stuff that had been beat down into the mud, before it all rotted. They could have left the potatoes and other root crops in the ground at least a little while longer without harm, unless they had a solid frost that froze the soil.
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