Cut the 2-by-6-inch wood planks to length with a circular saw. Cut each plank into two 3-foot lengths and two 2-foot lengths, for a total of four 3-foot and four 2-foot boards. Cut four 12-inch lengths from the 2-by-2 inch plank.
Lay two 3-foot lengths on their narrow side. Place a 2-foot length at either end of the 3-foot planks to form a box. Nail or screw the four corners together. Repeat with the remaining 2- and 3-foot boards to create a second box.
Stack the boxes on top of each other to create a 12-inch-deep box. Place a 12-inch 2-by-2 board in the interior of each corner.
Attach the 2-by-2 boards to the boxes with nails or screws. The 2-by-2 boards hold the two layers of the vegetable box together.
Cut a 2-by-3-foot rectangle from mesh hardware cloth using tin snips. Use a fine mesh cloth with openings no greater than ¼ inch.
Lay the hardware cloth over the bottom of the box. Staple or nail the cloth to the bottom of the box frame.
Set the box in a sunny location. Fill it with moist potting soil and plant your vegetable seedlings inside, following the spacing requirements on the plant labels.