Via Jacob A. Geller, the evidence is now in and it seems to suggest no, food deserts are not a real problem:

Here is more, and here is the study itself. If you look at the statistical tables, they’re pretty striking. Even where there is statistical significance — which is the exception to the rule — the size of the effect is so tiny, it’s like practically nothing. For example, on the margin, adding one full-service supermarket within a one-mile radius of your house is associated with an average BMI decrease in your neighborhood of .115. That is a difference of just one pound. (see back-of-the-envelope calculations here) So there is really no relationship, according to this one recent study of nearly 100,000 Californians, between the distance between your body and a full-service supermarket (or any other kind of food store), and whether or not you are obese. Distance, which is a proxy for access (the idea of a food desert is that the nearest supermarket, which has fresh produce, is distant), is for all practical purposes a non-factor.

Here is a good example:

For example, when you last ordered food at McDonald’s, did you even notice those ten salads on the menu? Did you order them? No, and me neither. And did you ask for a cup of water, which is free, instead of a soda? No again. (That’s my experience anyway, and that of millions of other Americans)

And an excellent parallel: