Arrange the four equal boards on their 2-foot sides to form a perfect square, with each board butting into one board while another butts into the side of it. Predrill four evenly spaced holes into each joint, with the holes entering the outside of one board and passing into the end of the other. Attach the boards together with 2-inch wood screws.
Position one of the square boards on top of the ottoman assembly and predrill one hole in each corner of the square board, down into the center of each side board. Attach the square board to the side boards using the 2-inch wood screws.
Cover the outside of the ottoman with tightly stretched fabric, covering the inside as well if you like. Staple the fabric in place on the bottom of the ottoman's surface and then again inside it, in the storage area.
Designate one side of the other square board as the top and one side as the bottom. Stretch fabric loosely over its top and fasten it underneath with staples on three sides of the board. Stuff as much cotton or polyfill batting onto the top of the square board as possible, until the surface and sides of the board resemble a cushion and are comfortable to the touch. Finish attaching the fabric by stretching it tight and stapling the final side on the underneath of the board.
Position the two hinges at the very top of one of the ottoman sides with the hinge portion pointing toward the outside edge of the ottoman. Attach the hinges to the ottoman with the 3/4-inch wood screws.
Place the lid of the ottoman against the hinges in the natural open position. Attach the lid to the hinges with the 3/4-inch wood screws. Close the lid to check the fit and make any adjustments necessary for an attractive final product.
Flip the ottoman upside down and add padded protective feet stickers to its base in all four corners and in the middle.