I'd actually say it's more of a intelligence problem in most cases than "poor food" problem. Plenty of people eat unhealthy food when they could do otherwise, simply because it's "easier".
In a lot of cases, people choose to eat poor food and not go outside to exercise so that they can stay healthy. I think that's an education problem primarily. And it's not entirely even that the food is the problem so much as the exercise. I've known a few families who are healthy through being active, but still eat rather poor food choices such as fast food, easy microwave dinners, etc. Although, they also have real home cooked meals somewhat often as well.
But for poorer folks, considerably poorer, we definitely have a problem with the low quality of food that is available to those people. But then... if the middle class (and upper class, I suppose, but I think they probably eat fairly "organically" being as money isn't an issue for them) would stop being, well, lazy, and just get back into the kitchen, then real food would start to become cheaper, allowing the lower class to have realistic healthy options.
And of course, with exercise, you can eat fairly poorly and still stay healthy.