One, or more of the owners of the land it accesses would have to petition the county for closure and probably list a reason like "it's impassable". In this example the land would appear to be owned by one person (entity??) on the north and south side. If so he could easily petition to have it legally closed because it accesses nobody else's property, especially public land.
I've been in attendance at a couple of different counties' commissioner meetings when this has come up. For the most part, I like that in all the examples I've seen, the county commissioners legitimately deliberate the closures. Not only do those section lines tend to go nowhere today, but the explore how it could in the future. They can also do temp closures, for things like gravel pits or oil development and put expirations on the closure.