Lay one piece of the 6-inch-long, 1-by-3-inch lumber flat on a work surface. Measure up 1 inch from one end and draw a 1 1/2-inch-long line perpendicular to the long side of the lumber, then draw a 1/2-inch tail on the line that is parallel to the long side of the lumber and faces down toward the end, so that it looks like a sideways “L.” Repeat on a second piece of 6-inch 1-by-3 lumber.
Cut a 3/8-inch-deep and 3/8-inch-wide groove along both the “L” shapes using a router with a 3/8-inch router bit attached. These grooves allow the paper towel rod to be inserted and removed as necessary.
Lay a piece of 12-inch 1-by-3 lumber flat on a work surface, and stand the two pieces of 6-inch-long, 1-by-3-inch lumber with the “L” shape grooves at either end. The two pieces of lumber should be on end, with the grooves at the top and mirroring each other.
Secure the two upright pieces of 6-inch lumber to the flat 12-inch piece with two 1 1/2-inch screws through the side of the two upright pieces into the ends of the flat 12-inch piece. Use a 1/8-inch drill bit.
Center the 12-inch piece of 1-by-3 with the upright pieces attached 3 inches in from one 24-inch edge of a piece of 12-by-24-inch 3/4-inch plywood. Secure the 12-inch plank to the plywood with a 1 1/2-inch screw through each corner of the 12-inch plank. Again, use a 1/8-inch drill bit.
Position two shelf brackets level on the garage wall, 16 to 20 inches apart. Secure the vertical arm of the brackets to the wall with 3-inch screws and screw anchors, use a 3/16-inch drill bit. Screw anchors will be enough for concrete or brick walls. If the garage wall is not concrete or brick and has studs behind it, screw the brackets into the studs, which are spaced 16 inches apart.
Sit the plywood centered between the brackets, resting on the bracket’s horizontal arm with the 12-inch and 6-inch pieces of 1-by-3 on the underside of the plywood. Secure the plywood shelf to the brackets with 1/2-inch screws through the screw holes in the horizontal arm of the brackets into the underside of the shelf.
Drill a centered hole in each end of a 11 1/2-inch-long piece of 1-inch dowel. The holes should be 1/2-inch deep and 1/4-inch wide, so use a 1/4-inch drill bit. Glue a piece of 1/4-inch dowel that is 1-inch long into each hole, pushing the dowel pegs into the holes so that only half of it protrudes at the ends of the larger dowel piece.
Slide the dowel pegs in the ends of the larger dowel into the “L” shaped groove on the 6-inch 1-by-3 pieces of lumber attached to the underside of the shelf. The pegs will rest in the short downward-facing tail of the “L” groove and hold the larger dowel in place.