Plastic has a lot going for it, of course; I'll say up front that my household once relied heavily on those big blue Rubbermaid tubs, and no basket of natural materials would have done the trick.
However, let's be humble and remember that "better living through chemistry" seems to have a dark side, too... BPA, anyone? (insert your own example here)
I was delighted to see some examples of Mi'kmaq birch bark basketry at the Nova Scotia Museum of Natural History in Halifax last summer. I have yet to have the pleasure of working with the kinds of birch bark that grow in the eastern forests and the northern regions, but I hear that it is sublime: more soft and pliable than any other basketry material. The baskets are sewn together with spruce roots, like a stiff fabric that holds its shape. (Working with split roots is another mystery I have yet to explore...)
Here are some other examples of birch bark forms and vessels; these are from The Kids Book of Aboriginal Peoples in Canada, written by Diane Silvey and illustrated by John Mantha. It's a book we found at the library and I would highly recommend it for kids under 11, for depth of info, clarity, balance and illustrations.
Look at how big those birch bark rice baskets are! They're almost as big as Rubbermaid tubs!