[kevingrahl]

I haven’t even thought about those wooden woven baskets!

But I suspect you‘d need two of them to equal the storage of a single standard plastic bag. And more than two baskets should be hard to carry.

[SllX]

Baskets come in different shapes and sizes, but if you aren’t picking up a lot, just enough to be a bit more than is easy to carry in your hands, or your mode of transportation precludes carrying your groceries in your hands, it could work, and double as your in-store shopping basket.