I hated H-E-B living in Austin (they are not created equal, and I would consider myself an "outspoken critic" of the chain) so I try to balance things out by visiting other grocery stores, and I believe I found the oldest continually operating grocery store in Texas, right there in Austin.
I've been in old stores before...like the Midtown Sears in Houston before it closed, but rarely do I see grocery stores older than 40 years. Unlike department stores which can stick around for years as long as the parent company or its successors don't give up on the property, grocery stores from the 1960s and 1970s tend to be smaller than their modern brethren, and thus tend to be lost in the shuffle, as they fall into the hands of non-grocers or third parties, especially if the area of town has changed since the grocery store was built.
In rare cases, though, that doesn't happen. In March 1953, Rylander's, a local Austin chain, opened a 13,000 square feet store at 2727 Exposition Boulevard. In the early 1970s, Cullum Companies out of Dallas purchased the chain and several years later rebranded the chain to their own name. In 1992, Cullum Companies was purchased by Houston-based Randall's Food Markets and renamed again in 1994. It received one expansion at some time that added a bakery/deli (likely recycled from an adjacent store space, bringing it to 17k square feet).
It probably isn't the longest-running continually-operating store in the country. Prior to 2007 when A&P exited New Orleans, the 1930 701 Royal Street location would've taken that crown (despite whatever damage it got from Hurricane Katrina)...and the current winner is probably the Broadway location of Fairway Market. But it is the longest in Texas, and that's what counts.
While it doesn't scream "1950s" on the inside (it has been updated over the years), the Randalls at Casis Shopping Center in Austin definitely is a relic of the past and worth checking out. There is a florist, but no pharmacy. The meat department exists but is an antiquated "ring bell for service" with very little display area for fresh meats. The bakery and deli exists, but they're both tiny as well.
Enjoy the pictures below, they were taken in May 2023.