Step 1: Think first
I personally needed it to fit behind the passenger seat in my car. This limited the dimensions, so I had to find a size that would work with different arrangements of beverage containers and ice containers. I chose an internal box size of 7W x 10.6L x 10H (inches). Different applications would require similar planning.
I considered a few materials for the insulation: FIP polyurethane foam inside a cardboard box, EPS (beaded polystyrene foam), and extruded polystyrene foam. Since the first two ideas suck and I already had a sheet of 1.5" pinkboard, I decided to use that, thus giving the box its absurd 3" thick walls.