When I was originally taught groups, one concrete example was the symmetry transformations of an object, like an equilateral triangle ($S_3$) or a rectangle ($V_4$). Usually you label or colour-code the corners so you can tell the difference between them.
One easy way to form a Cartesian product would be to have transformations of both object simultaneously. So an element might be "flip the rectangle horizontally and turn the triangle $120^\circ$". You can imagine the students with an actual rectangle and triangle and doing one action with one hand and one with the other.
A useful thing to do would be to find the order of such an element. So keep doing both simultaneously until both shapes come back to the original configuration. The number of times is the order. They should very easily come up with the idea that the order will be the greatest common multiple of the orders of the two individual transformations.