Arrange four 3/4-by-12-by-12-inch plywood boards on their long, thin sides. Position them in a square frame, with each board butting up to the next so that they form a 12 3/4-inch square frame. Position 1-inch metal L-brackets on the inside of each frame, with two evenly spaced brackets between each board. Mark the holes of each bracket and drill them out. Attach the brackets to the boards with 3/4-inch wood screws.
Set a 3/4-by-12 3/4-by-12 3/4-inch plywood board on the ground and put a square frame on top of it, so that the outside edges of the frame are even with the edges of the larger plywood board. Consider this board the rear board of the cabinet. Position two metal brackets between the rear board and each other board. Mark the bracket attachment holes and drill them out. Attach the brackets to the rear board and other boards with 3/4-inch wood screws.
Lower another 3/4-by-12-by-12-inch plywood board on top of the square frame evenly so that it is directly in the center of the frame. Position two side-mount cabinet hinges between this board, which is the door of the cabinet, and one of the side boards. Mark each of the hinge attachment holes. Drill out the holes and attach the hinge to the side wall and the door using 3/4-inch wood screws.
Put a magnetic 2-part latch in between the door and the opposing side of the side board with the hinges. Mark the latch holes in the door and side boards, and drill out the holes. Attach the latch with 3/4-inch wood screws.