Located on Ocracoke (sounds like Okra Coke) Island. The island is reached from Hatteras by a free car ferry. It is about 16 miles in length with only the last few miles comprising the town of Ocracoke. You can certainly find stretches of private beach along these shores along with plenty of 4x4 beach trails. The beach front is part of Cape Hatteras National Seashore. There is a section fenced off for the wild horse to roam, although I only saw the few in a pen. By law, the wild horse of the Outer Banks are not permitted to roam and must be kept penned. If you go to this little island during the season, expect long waits for the ferry and a caveat, the shops appear to close up early but most of the dining was open.
If I weren't so tired, I'm sure this little travel log would be more coherent.