In NC the most remote is 11 miles, the summit of Mt Sequoyah to the nearest road (in the GSMNP). I think there are places as remote in the the Alligator River Refuge or Dare county bombing range of eastern NC. Although there are roads many are gated and not accessed by private vehicle.
This may note be remote in terms of distance from roads and all that, but the upper section of Mud Creek Canyon on Mount Shasta is pretty inaccessible. I doubt many people, if anyone at all, has ventured in there. It is pretty inhospitable: