Lay a plywood sheet flat on a hard surface. Draw three lines across the sheet that are parallel to the longer side. Space each 4 inches apart and with the two outside lines 1½ inches away from the edges. Draw another three lines that are perpendicular to the first three that are spaced 9 inches apart and with the two outside lines 1 inch away from the shorter side. Repeat this process on the other sheet of plywood.
Drill a 1/2-inch-deep, 3/4-inch-diameter hole where each of the lines intersect on both sheets of plywood. Erase the remaining pencil lines on the plywood.
Position the plywood pieces 30 inches apart with the holes facing each other. Insert the nine dowels into the holes. Screw through the plywood and into the dowels using the wood screws.
Paint or stain the shoe rack to protect the wood.