Do you know of any retail stores going out of business near you? You may well be able to pick up one of their shopping carts for pennies on the dollar at a fixtures sale.
Even more good ideas! If my dad was still alive with all of his tools, I’d commission him to whip one up in the shop. I just basically want a body to hold the groceries, big rubber wheels to absorb the shocks of driving it, and an ergonomic handle that I can push in front of me without stooping and having my arm angled out behind me. This thing totally should be invented.
A backpack?
The traditional Dutch answer to this problem is a sturdy city bicycle with side bags, like this http://fotos.marktplaats.nl/kopen/4/2c/3cx+OZCPBt1YLrY8e6Ne5Q==.jpg or this http://nadekomma.files.wordpress.com/2007/09/dsc00026.jpg
Two 50 litre bags will hold 5 days worth of shopping, not counting bulky groceries like cat litter, tolet paper and a crate of Coke or beer 
…or the requirements of teenage boys :eek:
When I was without a car I had similar issues.
Like many of the doper’s here, I gave up and bought a child’s red wagon, second hand, very reasonable, with detachable sides, rubber wheels. A wagon is actually very maneuverable.
The only alteration I had to make was a cloth loop handle, added to make it the exact right length for me when pulling. It wasn’t ideal, but it worked. Better than I thought it would. The wagon proved to be just the thing for several other, large heavy things, that could never go in a grocery cart.
Of course, they didn’t have some of the great suggestions I’ve seen in this thread.
I gave it away to a small child a few years ago now, but it did work quite handily for whatever I needed.
But what do you do with the wagon while you’re in the market?
I have a granny cart which slips in between the handle and body of the shopping cart while I’m in the store. I would love something sturdier with better wheels.
I always locked mine, by the handle to the bike rack, with a bike lock.
The other thing I liked about it was it didn’t make a hideous noise and draw attention. I despise all childrens toys made with plastic wheels, they are abominations! I cannot imagine being a young child dragged around and being subjected to that racket for a whole afternoon!
I think this http://www.faticahardware.com/grocerycartsstepstools.html is what you’re looking for. Look down the page, there are 18 different models.
This http://premiercarts.com/ may be even better.
Caveat: Premier Carts only sells the folding carts and the wagon carts to individuals, per their FAQ.