I’ve been to a teahouse in Kyoto that’s been open since the 15th century. It’s, er, not exactly in my area though (sorry).
Red Knapps Dairy Bar, Rochester, MI. I drive by it every day I go to work. I only have seen it since the 80s, but it presumably hasn’t changed since it opened in the 60s, and I like to think it has continuously played the exact same set of music.
Pellici’s, a cafe in Bethnal Green. There are lots of businesses that have been around for longer - this cafe opened in 1900 - but this one is unusual in that the same family still run it. They’ve never turned it into a chain, they just run a very small, crowded cafe and the food is fantastic. The service is even better, nonna in the kitchen cooking the food, grandkids serving, and genuine locals are still the major clientele. I took my ex’s Dad when he was feeling a bit down about the world, and they chatted to him loads about where he was from and gave him a free bread pudding as we left. There was nothing obvious about him to say he was sad, they were just lovely, lovely people.
They still close every August like old Italian restaurants used to in the UK (and some in Italy still do).
The decor is gorgeous, mostly from the 1940s.
'You Can’t Get Nothing Like This No More’: How London's Oldest Family-Run Café Beat Gentrification - Everything here is true. There is no way to overstate how wonderful this place is. It makes you feel better about the world.
Isn’t there some hotel in Japan that’s been in operation continuously for something like a thousand years?
Ok, also the clowns. 
Based on your location I thought you were going to nominate the Tadich Grill in San Francisco, which claims to be California’s oldest restaurant, founded in 1849 at the start of the Gold Rush. Although it looks like they’ve “only” been in their current location since 1967.
I work for a plumbing supply house that’s had only two owners since 1929.
Mickie’s Dairy Bar (a greasy spoon across the street from the University of Wisconsin) has been serving huge griddlecakes, five-pound scramblers, and malts for breakfast since 1946.
They still have an old 50s menu on the wall.
Waikiki has changed greatly from our student days in the early 1990s, but Roughage Natural Foods is still in the same dinky little shop it’s always been in on Kuhio Avenue. I’ve never been inside, but I know it’s very popular.
Yeah, but it’s 6 miles away, much too far to drive.
Locally, we have Shanghai Chop Suey, from 1949. It’s your typical Chinese-Canadian place, such as you might find anywhere on the Prairies. Sweet-and-sour chicken balls, chicken fried rice, beef-and-broccoli, and so on. And of course, chop suey.
Though other places in town have done Chinese cuisine well and better in the years since (Mimi’s comes to mind), the Shanghai does well, business-wise. It’s the tradition that keeps it going, I guess; though it is tasty when you want Canadian-Chinese cuisine.
Fuller & Son hardware has been in Little Rock at least since the sixties. They have 4 locations. I think the downtown location is the oldest.
Kaufman Hardware is a similar age.
That’s the oldest I can think of.
Sears in N Little Rock dates to the late sixties when McCain Mall was built.
I checked, McCain Mall opened April 1973. Sears is one of their anchor stores.
Meh, there are loads of those. There are loads of businesses near me, like within a mile from me, that have been in the same place for centuries, but to me it’s more impressive if the company is still also in the same family.
James Smith and Sons Umbrellas might count. The location is from 1857, the decor is from 1865, and it seems to be part-run by members of the original family. It’s a really beautiful building to pass by on the bus, and a friend of mine went there for a walking stick because apparently they are genuinely better.
Placerville Hardware, “The Oldest Hardware Store West of the Mississippi” was founded in 1854, and has been in continuous operation in the same building since 1856.