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.