Although I obviously don’t remember an era with no cars, I do remember one with few cars.
In 1946 when I was nine, my family moved to a lower middle class neighborhood in West Philadelphia where relatively few people had cars. We got our first car in 1953. There were three small one man (sometimes with his wife’s help) grocery stores within about 100 yards (90 meters) of our house and a small counter-service A&P about 400 yards away. We went at least once a day, often more often, to one of the nearby groceries, rarely to the A&P. My father worked about a mile away and usually walked, although there was a bus he could take. My mother (who suffered from agoraphobia and really disliked leaving the house) would usually order clothes from department stores, who delivered, and it was my job to take a trolley (that is, tram) downtown to return the ones she didn’t want.
When I got home from school, and all day during the summer, I went out to play on the street until dinner or darkness, whichever came first. The streets carried few cars to interrupt our games and, even more importantly, hardly ever were cars parked on the street. By 1950, there were more and more cars and this way of life was ending.
Oh, yes, there were trucks, often horse-drawn, that came down the street regularly. Fresh fish on Fridays, fresh vegetables during the summer (actually, we had a green-grocer a couple blocks away, but our regular grocers were not equipped to handle fresh produce, although they did sell potatoes and carrots), milk and other dairy products, and, oh yes, blocks of ice. Although we had a fridge, my grandmother a couple blocks away, had an ice-box in my memory. Of course, the horses delivered some unwanted things too.
But although this could always be done with horse-drawn vehicles, it was much more efficient with motorized trucks. Aside from package deliver, there was trash (dry), and a separate garbage (wet) pickup and, nastiest of all, ashes. Until the late 40s, nearly everybody heated with coal (anthracite, as a matter of fact, Pennsylvania being the main source in the entire world) and those ashes were heavy. The pickup was choreographed with big trucks that carried 3 containers and had a crane and one container trucks that did the actual pickup. When they were full, they went to one of the big trucks that used its crane to exchange an empty container with a full one, and when it got full, it drove off to be replaced by one with three empties. I wonder when that ended. We got oil heat in 1950.
The OP asked about car-free and we essentially lived a car-free existence. But vehicle-free is another matter entirely. That would mean not only no trucks, but also no buses or trains. I wouldn’t want to live there. But I live nearly car-free today. Aside from distance trips, I drive to the supermarket once a week and downtown to concerts maybe every other week and otherwise my car sits in the driveway. I drove to Ottawa 2 months ago, filled the tank on the way to Ottawa, drove there and home (Montreal) and still have not bought gas since. I will have to buy some within the next couple weeks though.