On Aquinnah's North Shore you'll find Lobsterville Beach across the channel from Menemsha, and stretching for miles up to Dogfish Bar. This stretch of beach is very secluded. There are sometimes closures on the beach to protect migratory bird nests. Parking is extremely limited but there will always be plenty of room to stretch your legs.
Lobsterville Beach's West Basin can also be accessed by the bike taxi in Menemsha, this is a privately run service that runs from Menemsha, to West Basin boat launch.
Once you are on Lobsterville Beach there are no facilities or amenities of any kind so plan accordingly.