Absolutely totally completely possible. We live in NYC, just over a mile from a costco, we walk together and make a Saturday / Sunday afternoon of it. After work is even better since it's far less busy, but that's tougher with the hours we work and it being so dark at night in the winter.
Most common option for us: Bring lots of reusable shopping bags + granny cart. I typically carry 2 reusable shopping bags, and my fiancee does the granny cart + a smaller reusable bag. We can get around $250-$350 worth of stuff this way without a problem.
Second option: Bike with a trailer. I am looking into getting a bike trailer, because this will be way quicker and easier. I've seen people there with trailers, so it's definitely possible.
Third option: Rent a cargo van for cheap ($20-$30 for a few hours) and do a huge shop with a friend who is also a carless member (neighbor, etc). Requires more coordination, but for those few hours, it's nice to have a vehicle and not have to even think about the weight/size of items you're picking up. We have done this once and it is good particularly if you have a trip where you're doing a major restock: large bags of flour, rice, things of olive oil, cleaning supplies like laundry detergent, etc. I think the last time we rented one in NYC it was about $35, with all taxes / fees.