If you buy solid cloth bags and put them through the washing machine regularly then they're fine. It's what we do.
With the more solid plastic bags that you mention, if contaminated you can give them a wash out in the shower with washing up liquid and warm water.
People managed to go shopping without plastic bags for many centuries. Woven baskets were one of the more favoured shopping bags - and it is better for your back to carry a basket over your arm tucked against your hip than bags dangling on your fingers and cutting circulation as well. (I am talking the sort of oval baskets with a bit of an indent near the handle that will curve round your hip.)
Where we are putting shopping straight in the car, we put it unbagged back in the trolley and load it into cardboard boxes in the boot. For cold things there is a cool box. For damp things you can have a re-use it many times robust plastic box.
The article isn't just about shopping bags though, it is also all the fast food containers, milk bottles, plastic cutlery etc etc.
There is a growing number of "fill your own" shops, where with certain products you help yourself into a container you've bought. Marks and Spencer in the UK has embraced this on some lines - and they are a big name, more upmarket store.
Marks & Spencer has extended its packaging-free refillable grocery concept Fill Your Own to eight additional stores – bringing the offering to 11 stores across the country.
www.retailgazette.co.uk
When I was a kid, food went into paper bags, not plastic. The classic for fish and chips was it was handed over in a folded newspaper. Don't see why you couldn't bring your own washable tub of choice to a takeaway and they measure in whatever it is you are buying.
There was a story about the Duke of Wellington - the general - about how at a dinner part on campaign the officers all sat down and found no cutlery. The Duke then gifted each one of them with a set of travelling cutlery and lectured them on preparedness.