I always used to walk it home in a cart.. everything was close by where I lived, so it wasn't too far and I preferred walking anyway. On bad weather days I would call a cab.
What me and my family use to do when we didn't have a car is we walked! We would take the cart with us and walk home with our groceries in the cart.
I have thought about this also. I used to be in a similar situation. Lucky for me, I lived a block from the nearest grocery store. I would take their cart home, then have to walk it back to the store. As, in most places, it is considered theft to take the cart home. They have shopping carts available online at sites like Amazon. They are reasonably priced. You just walk to the store pulling one of these "old lady" (no offense intended to anyone) carts and then put the heavy items on the bottom. Then you will not need to walk back to the store and return the cart.
I have supermarkets near my flat so if it's not too many things I can walk with them, other than that I must take the car. I can't imagine how people without car manage, possibly they have to shop more often.
I have been renting a Zip Car to get my chores done that I can not get done on public transportation, the bus in my area runs once an hour and not at night or weekends which is very limiting. Renting the Zip Car for a few hours a few times a month can get the job done to shop at out of reach places, pick up more that a quick trip on a bus will allow and even do a quick job to pick up a few dollars. Renting Zip Car is not really cheaper than the normal rental places but you do not have to have your own insurance for the automobile when you rent from the normal places like Budget, where the insurance is very expensive.
I don't drive, so I use public transport and if I walk I take a rolling bag with me and I'm fine. People with cars get used to it, but when you don't have transport you can survive. There is also free delivery from many supermarkets so you may not get so many bargains, but it's still very useful.
If I lived close enough I can easily see myself getting a little cart and making the walk. I did that years ago, in fact, when I lived in Knoxville, TN and my husband most times had the car. Now days I always have access to a car. Or truck, in my case; but often opt for taking my scooter instead. I have a little Honda Reflex with storage under the seat and a nice sized top box. You wouldn't believe the things I can carry on that scooter. It will hold a good sized watermelon under the seat alone. Plus it's fun and it's cheap to ride.