I've got a 12" cube cooler and my wife and older two kids and I can pack lunch for everyone in there pretty easily. Might need a second one for drinks but I feel like it's plenty of space for a single person.

That said, if you go multiple times or go on really hot days those refill cups are terrific and alleviate needing to bring water. Got them with our season passes last year and it was much better than our usual lugging around a dozen water bottles in the stroller.