Chilmark is a town on the west coast of the island of Martha's Vineyard which includes the fishing village of Menemsha. Chilmark was once home to a large number of deaf people (1 in 25 people was deaf) who developed Martha's Vineyard Sign Language. Today Chilmark has among the highest property values of any Massachusetts town.
Lucy Vincent Beach. Located off of South Rd in Chilmark (listed as Lucy Vincent Rd on some maps, it is a dirt road just before the big bend at Windy Gates Rd), this is a private beach for town residents  during the summer. Nudity is allowed in the area that is more than 1,500 feet from the entrance. It is open to the public during the off-season, October 1 through May 31. Quite popular and scenic as it sits on the border of the eel-grass dune stretches to the east and the beginning of the clay cliffs to the west.
Menemsha Beach. Public. At the end of Basin Road, just past the docks. Some parking available but gets full on weekends. Pebble bottom, cooler, clearer water than the south beaches. Protected by jetties on either side so waves are minimal. Lifeguards during daytime. Popular sun-set spot towards evening.
Larsen's Fish Market, Basin Rd, Menemsha, Phone: +1 508-645-2680. Long-time Menemsha fish purveyors. Not actually a restaurant, primarily fresh-caught seafood on ice. They do stuffed quahogs and the like towards the back.
Menemsha Galley, 515 North Rd, Menemsha, Phone: +1 508-645-9819. Consistent fast food, burgers, shakes, lobster rolls. Outside ordering at the front window, a couple of picnic tables for seating on the porch around back.
Theo's at The Inn at Blueberry Hill, 74 North Rd., Phone: +1 508-645-3322, .
The Beach Plum Inn & Restaurant, 50 Beach Plum Ln., Menemsha, Phone: +1 508-645-9454, .
At the Cornerway
The Bite, 29 Basin Rd, Menemsha, Phone: +1 508-645-9239. . Mainly fried seafood & chips, good clams. Just a shack, no indoor seating.