Veggies and nuts are mostly grown at massive industrial operations in California with desert sunshine and irrigation systems. It's definitely a scalable operation not much different from corn and soy.
Not to nit-pick, but carrots keep really well even at room temperature. That was the whole point of root cellars. Veggies store fine. Lots of "fresh" stuff at the store routinely comes out of 9 months in cold storage, and that's without the complex chemical processing and packaging applied to the junk food. Then there's freezing and pickling.
The prevalence of crap in the US diet comes from a deliberate government plan to pump out more cheap calories. It might have made some sense in a Marshall plan world, but we're still stuck with it. That's the story as I understand it.