Neversink General Store Share: Convenience Stores Neversink General Store Visit Website Request Info 4 Shumway Road Neversink, NY 12765 (845) 985-2076 (845) 985-2013 (fax) Facebook Hours: Monday-Wednesday: 6:00am-9:00pmThursday-Saturday: 6:00am-10:00pmSunday: 6:00am-8:00pm About Map About The Neversink General Store - Grocery and Gas StationGreat homemade food, Local products, Free WIFI, and so much more! Not just a convenience store, we are a community! What's Nearby? Distance: 5 10 15 20 25 50 mi.