From any of the 5 park entrances, the furthest distance you'd have to travel without passing a park concession / food service is 29 miles. Interior "foodless" intervals are shorter.

Turns out eating figures greatly into any modern tourist's daily schedule. See if you can find a dot on this map where you can't get a burger and fries...

http://www.nps.gov/archive/yell/inte...emap/index.htm