Rather than using the standard store-bought baskets or fabric boxes to keep my stuff in, I wanted to make some myself with my own fabrics. After some attempts that required turning the whole thing inside out including cardboard (quite impossible) and seams showing on the inside of the box, I came up with this final version. No seams are showing at all, and it's super quick and easy!

With this instructable you can make your own, out of some cardboard and any (non-stretch) fabric you like. The following steps take you through the process. The basic idea is to make a 'box' out of fabric, with sides twice the desired height. Then place cardboard pieces inside this box, and fold the leftover fabric in over the cardboard. A piece of cardboard at the bottom wrapped in fabric holds everything in place. No glue is needed since the stuff you keep in the box will press down the bottom.

Additional note: instead of cutting cardboard bottom and sides, you could also use an existing cardboard box. All you need then is an additional bottom piece to cover up the inside.

Step 1: Measure

First, you'll need to measure just how big you want the box to be. In my case, the box is meant to go in a small cabinet with shelves of about 15 cm deep. I decided to make the box 15 x 15 cm wide, with a height of 10 cm.

Step 2: Materials

To make the box, you'll need cardboard, fabric and thread. The amounts depend on the size of the box. For a box of 15 x 15 cm wide and 10 cm high, you'll need:

30 x 40 cm of cardboard (about 2 mm thick)
40 x 50 cm of fabric, non-stretch such as a basic cotton fabric
thread

Step 3: Cutting the Cardboard

Draw and cut the following pieces of cardboard:

2 bottom pieces, in my case 15 x 15 cm
4 side pieces, in my case 15 x 10 cm

Step 4: Cutting the Fabric

Lay out the fabric with the wrong side (the back side) facing up. In this step, you'll put the cardboard pieces on the fabric and trace around them with a pen (a regular ballpoint pen will work), to mark where you need to sew in the next step. When cutting, add about 1.5 cm around the markings for the seams. You'll end up with three pieces of fabric, as follows:

Lay out the four sides of the box next to each other (as they would be next to each other in the box), with a small (2 mm) space between them to allow for the fabric to bend around the edges. Trace around each piece of cardboard with a pen. In the small space between two pieces there should be only one line marking. Cut out around the whole of these four pieces with an extra 1.5 cm of seam allowance.
Lay out each bottom piece separately, mark around it and cut with a seam allowance of 1.5 cm.

Step 5: Sewing

In this step you'll sew together a fabric 'box' that has twice the desired height. You can sew with a sewing machine or by hand. Make the box inside out, and then turn to have the good side (front side) of the fabric show.

Take one of the bottom pieces. Lay the long side piece on the bottom piece, with the good sides facing each other.
Make sure the markings of one side of the box match exactly: the line markings on one side of the bottom piece and the first part of the side piece should match. Pin this side and sew.
Now go around the corner: make the next side match and pin. This is shown in the first image.
Keep pinning and sewing each edge until you reach the beginning.
Finally, sew the two ends of the side piece together. The result will be a 'box' (more or less), as shown in the second image.
Turn the whole thing inside out.

Step 6: Putting the Cardboard Pieces In

Finally, put the cardboard pieces in and fold the fabric in as follows:

Put one cardboard bottom piece on the bottom of your fabric 'box'.
Put all four cardboard side pieces in. This can be a bit tricky at first, but once you have them all, the box should be able to stand up on it's own (first image).
Fold the fabric back in over the cardboard sides (second image) and tuck in the corners.
Take the second bottom piece (third image) and wrap the fabric around the edges. Secure with some glue or tape. Technically, the fabric will stay in place by itself, but for putting the bottom piece in, it's easiest when the fabric doesn't move too much.
Finally, put the second bottom piece on the bottom of the box, with the fabric covered side facing up. This bottom piece will hold down the fabric folded over the sides, and hide all seams and additional fabric. You can glue it to the other bottom piece, but that's not necessary: by putting stuff in the box, the bottom will be secured and will keep everything in place.