If you do not have a big enough piece of cardboard to follow the diagram shown, cut like so in the picture to the right, and cut a separate square. After this is cut, use masking tape and a stapler to form into a box. Make sure everything is secure, and then reinforce with more tape and staples. In the end, your box should look something like the picture below.
To make the fabric cover, follow the diagram to the left, pink being the patterned fabric, and blue being the lining. For the patterned fabric, sew a fabric cube in a similar way to the cardboard cube, so that it fits snugly around the cardboard. For the liner, cut and sew your lining in a way leaving the top and bottom open. Let the liner be a little longer than the actual cube. Sew one open end of the liner cube to the open end of the patterned cube, and then push the cardboard cube inside. Then, push the liner inside of the cardboard cube. Take a square piece of foam core board, cover it in liner, and push it inside the cube. This will pull the excess liner and the liner on the sides of the cube taut.
To make fabric drawers, I cut Nike shoe boxes to size, and stapled when needed, then did the same thing with the fabric covers.