get somebody else to drive you there once a week, either for friendship or for money. There are taxis, and then there is the “domestic gigs” section of Craigslist where some people do in fact ask for people with cars to help with shopping.
I spent about a year without a car in college. Fortunately, the grocery store was only about a mile away, the laundromat half that distance and downtown (small college town) a couple of blocks. I made quite possibly the best transportation investment of my life: a Radio Flyer wagon. Nearly thirty years later, I still have it.
When I would walk to the supermarket with my little red wagon, I’d take it into the store with me and use it instead of a grocery cart. You’d be amazed at the amount of groceries that will fit in one of those things.
I do have a car, but I only use it for shopping when I need a whole bunch of heavy things. I normally walk every other day or so to one of three grocery stores within walking distance of me, and take the stuff home in mylittle cart. I need the exercise, anyway, so I buy small amounts quite frequently.
If I had to feed a family, too, I’d need a bigger cart. 
I don’t have a car. I buy small amounts of stuff on the way home from work - the supermarket is right at the station. If I’m doing a bigger shop on the weekend, I just carry it all home. The walk is less than 10 minutes.
I lived for over five years without a car in Egypt, and just walked to the store and carried groceries home. It was good exercise, and thinking about our wonderfully car-free lifestyle is making me miss Cairo!
If we were buying a lot of heavy stuff, we could have the store deliver - that is a standard service in Egypt. Elsewhere, I suppose I’d just take a taxi.
Have a car now (seemed like a good idea when the first kid showed up) but got by fine without one for [mumble] years without finding it inconvenient.
In fact, I often actually miss the way that I shopped when I was carless.
The main thing is that I’d select my residence carefully - I would make sure that a good variety of shops was convenient to get to - and ideally situated between home and work - which would of course also be fairly proximate to where I lived.
Rather than shopping for a week, I tended to make more frequent little micro-shopping trips to little markets within walking distance, usually on the way home from work.
I miss this because the quality and freshness of my produce and variety of available items was* much* better than I end up when I make weekly trips to a super market.
Now that we have a car, other factors have weighed more heavily than making sure we were ideally situated adjacent to a wide array of shops and services. Sure, it’s easy to do the one-stop thing and load up once a week - but I was much happier with the quality of my food when it was more “Today I’ll stop in quickly at the Italian deli,” and “I’ll just swing by the farmers’ market and pick up some nice fresh produce,” and “Hey, I’ll just duck into the bakery for five minutes on the way home.” “Oh, I should run out to the Vietnamese grocer and pick up a few essentials.”
I bring a backpack to the supermarket with me. When I’ve lived elsewhere I’d get a lift from a family member or take a taxi when I was getting a huge shop up.
I shop every couple of days, for a few things, and I live about 4 minutes walk from Tesco. I’m in the UK, though, and almost everyone will live close to a supermarket, if not walking distance then a short bus or taxi ride.
This, although I have a car - we’re in the same city. (Although probably not the same suburb). I just don’t have time to drive in and shop with uni and work, so we get Coles or Woolworths to drop it off. $9 if you plan it right. It’s less than what my time is worth to go. It’s worth that to me to not have to spend an hour and a half at crowded supermarkets on weekends with other people (and their spawn) in the way of my perfectly planned Shopping Experience.
When we didn’t have a car, it was either delivery, or walk to the shops and bus or taxi home. I’d get one big shop in every week with some sort of transport home, and thent top up at the local shops through the week with meat and veg for the next day or two.
ETA: I also had a shopping trolley, which was very daggy but very useful!
In my suburb I’ve found the best time to go shopping is at about 0800 on a Friday morning. All the dole bludgers are sleeping off their hangovers after spending yesterday’s pension on woody and cokes, and it’s just after the little ladies are finished their crack-o-dawn shop, so Coles is just about deserted at that hour 
I have brought my big backpacking pack to the store with me. I usually refer to this as going “Urban Sherpa.” Anything under fifty pounds is not too big a deal.
I find, well, about now really, 2AM, the best time to go. I work shifts so it’s usually convenient, and Tesco is empty, and I can shop in ten minutes.
I pick up groceries a couple times a week on the way home. I always carry at least one reusable grocery bag in my satchel. There’s a grocery store right by work, and the El station is right next to that, so four stops and I’m home.
Every couple weeks I use my 2-wheeled cart to get litter and canned cat food at Target, then take the bus home so I don’t have to haul the 40+ pound cart up the El stairs. My cart fits between seats on the bus, so I don’t take up any aisle space. That really irks me. If I don’t feel like going to Target, there’s a froufrou pet boutique I can walk to with my cart to pick up litter, but it’s a lot more expensive there. Only about a 7 minute walk, though.
It seems to me a majority of heavy or bulky stuff people buy is cases of soda and bottled water. Since I use neither of these, there’s just no need for more than a shoulder bag of stuff at a time. And it’s true, I think, that city people with markets nearby just don’t stock up. There’s no need. I have two organic stores, a produce market, and two large grocers (Dominick’s and Aldi) all within 5 to 15 minutes walk away. Then there are the farmer’s markets, one a block away and the other three blocks.
People with families, the majority (of ones I know) are double income and have cars. One family of four uses two wheelie carts and walks, that’s the only family I know without a car. I’ve seen people use a double stroller with one kid in it, and the second seat and under carriage filled with groceries on the bus.
If I need to pick up furniture or something equally heavy or bulky, I have iGo and just grab a car that’s appropriately sized for my needs. I also grab a car if one of my cats needs to go to the vet. I could use CTA, but the trip’s obviously faster by car and less stressful for them.
No 24 hour supermarkets around these parts. Besides, 2am you’d just get drunk-ass dole bludgers puking in the aisles if that were the case.
I have two grocery stores and a gourmet natural foods store, and a pharmacy and a Japanese specialty foods store, within a block of my house. I walk past all but one of them on my way home. If we need something, I buy it (or my husband does). I don’t do a massive weekend shopping.
If for some reason I need tons of bulky stuff, I have a little wheeled cart. Everyone has one in the neighborhood.
ETA: if something larger is needed, there’s always Zipcar.
Edited again - every place in NYC delivers. Aside from Fresh Direct, the grocery stores themselves deliver
One of my cultural blind spots when I moved to Australia was the inability to understand that there are no 24 hour supermarkets (at least, not around where I live, and I suspect nowhere.) 2am used to be my preferred time to do anything, but I worked shifts then, too, or at home with no set hours.
I still get caught out (It’s closed? WTF?) but it’s less jarring now.
Surprisingly few of them when I go, although I don’t go on a friday night. Usually because I’m out being drunk somewhere else though…
We have a cart we can take to the local grocery store which is about a block from our apartment or to the huge supermarket 5 blocks away. Once a month we do a bulk order from freshdirect.com so we get the really heavy stuff like cat litter delivered.
I need to get a good pack! Carrying things with your arms or even with a shoulder-strap (my current work-shop bag) is so much more difficult.
When I knew I’d be without a car for a week out here in the 'burbs, I stocked up ahead of time and figured, fine, all set. And of course I forgot milk, baking ingredients, and coffee. So I set out for the gas station convenience store up the road, risking my LIFE walking on the shoulder of the road for 20 minutes, and paying through the nose when I got there. And decided a bottle of soda would be nice. What a dope! I carried it in a bag, I swear my arms were twice as long as when I left the house.