Find and mark the studs in the wall using the stud finder. Hold a 24-inch 2-by-4 vertically against each stud, with the flat side against the wall, set at the desired height. Drill a hole every 6 inches through the board and into the stud. Attach the board to the wall with 3-inch wood screws. Repeat for all boards.
Stretch a tape measure across each 2-by-4, spanning the space between the walls. Cut four 2-by-4s to this length with a saw. Position a board on top of the vertical boards, spanning them. Drill two holes down through each of these horizontal boards and into the end of the vertical boards. Repeat with 2-by-4s underneath the vertical boards.
Measure the distance of the area above the 2-by-4s, both the distance between the walls and the distance from the front of the cabinet to the back. Transfer these measurements to a plywood board and cut it out with a saw. Cut out another plywood board to these measurements as well.
Position a plywood board on the top of the cabinet frame. Drill 2-inch wood screws through the plywood into the frame every 6 inches. Repeat with a plywood board on the bottom of the frame.
Measure the height and width of the cabinet frame. Cut out two plywood boards that are the same height and 1/4 inch less than half of the width of the frame.
Position these two plywood boards against the front of the cabinet frame. Slide two cabinet hinges between the frame and these boards, which are the door boards. Drill out holes for the hinge holes in the door and the frame and attach them in place with 3/4-inch wood screws.
Position a three-piece center latch in the middle of the frame on the top board. Mark the attachment holes on the doors and the top frame board. Drill out the holes and attach the latch with 3/4-inch wood screws.