Meat, Eggs and Milk in the Ice Chest
One advantage of car camping over backcountry camping is you can afford to take along an ice chest. So long as you keep the chest stocked with ice, you can take any fresh food you would otherwise store in your refrigerator. The main limitations of the ice chest are space and keeping it stocked with ice. Reserve the ice chest for relishes, fresh meat, eggs and milk.
Bread
Fresh bread can last for several days at the campground. The rule here is the same as in your pantry: keep the bread in a plastic bag in a cool, dry place with the minimum of air in the bag. This extends the period before the bread becomes stale. After becoming stale, the bread still has uses. Crumbled it and served as an ingredient in camp stews or use as batter for a fish fry.
Cheese
In the days before refrigeration, cheese was a way of preserving animal protein for travel. In the campground context, a hard cheese with a thick rind or a wax coating will last several days or even weeks if kept dry and out of the sun. Mold may form around the outer edges, but simply cut this away and eat the rest.
Vegetables
Root vegetables will last for weeks at a campground if kept in a cool, dark, dry place. This category includes potatoes, onions and carrots, as well as seasonings such as garlic and ginger. Ears of corn left in the husk are not quite as robust, but they will still last several days without refrigeration if kept out of the sun.
Fruits
Suitable fruits for the campground include lemons, limes, oranges and apples. If you are shopping for a long camping trip, rather than simply raiding your kitchen stocks, seek out and purchase irradiated fruit. The irradiation eliminates most the rot-inducing bacteria on the skin of the fruits, greatly prolonging both their shelf life and their life at the campground.