It’s 53 miles from Cody to the East Entrance in Yellowstone.  However, once you get into Yellowstone National Park, you don’t run into any services until you get to Fishing Bridge, which is another 27 miles.  So don’t expect to just get in the entrance to the Park and be greeted by a friendly Yogi-type bear… you’ll have a ways to go before you can stop for a meal and a tank of gas!