Absolutely a reality. I live somewhat in the middle of one. It's a decent hike to a grocery store, but the corner stores are only 3-4 blocks away, and they will gladly take food stamps. I regularly use public transportation and/or walk to the store, so I know the routes and know what I pain it would be for some of my neighbors to get to a proper grocery store or farmer's market. Especially with kids. "A couple of miles" is a long way when you don't have a car. You can't make a trip to the market every 4-5 days for fresh produce. It's a lot easier to find someone to watch your kids, make one trip to Walmart, and come back with enough cheap frozen food to last a while.
Now, as to the cause of food deserts, whether they contribute significantly to obesity, and whether or not there is enough demand for fresh meat/produce to support the supply within those areas, those still seem to be unanswered questions. Purely anecdotal, but I can say that some of my neighbors have been extremely willing to take any bumper crops we've seen from our garden so far. There is a new farmer's market about to open a few blocks from my house and I believe they plan on taking food stamps, so we'll see what the demand looks like there. I think there's a much larger culture and education issue at play though, and folks I know who work for charities or non-profits in the field say that to be the most important issue. Simply providing food sources won't change peoples' habits.