You can actually just look to what people were using before plastic became ubiquitous: wicker, cotton, burlap

Unfortunately if you live in the US you're starting out at about $50 instead of $5. I think that's a little bit of a HCOL price point, though. I picked up a burlap shopping bag in India earlier this year for about the equivalent of five cents, that any store here would have charged a couple of bucks for.