These woven storage boxes (ஓலை பெட்டி) were once used in India to store a lot of things. I even remember seeing wooden cookery set (சொப்பு சாமான்) and dolls (மரபாச்சி பொம்மைகள்) packed in these and sold in village fairs.
I was pleasantly surprised when I found out that Flipkart is selling wooden cooking toys. Am ordering one when I go to India next time. Not sure if I am buying it for little missy or for myself. They have 2 sets. I am adding pictures of those two sets below.
Procedure for making woven basket / box.