There are a number of dimensions the box could have with the desired volume, but the smallest difference in dimensions of an enclosure typically yields the least amount of the enclosing dimension for the most of the enclosed. e.g. what is the minimum length of fence needed to enclose a field of 40 ft^2. The answer would be 80ft (20 ft on each side).
In your box, we could get 2 ft on each side. That would lead to sides with an area of 4 ft^2. Since a closed box is a cube, it will have 6 faces, each 4 ft^2, that will give a total surface area of 24 ft^2.