But I doubt Barclays is currently upholding contracts signed 200 years ago.

When worrying about surviving for hundreds of years, you need to compare with the data about how many have existed for hundreds of years and then folded anyway.

Note that the above list includes a bunch of "national banks" which were founded at the peak of "national movement" in 1800s (as "nations" started forming "national countries") — with many of them, you can't really leave a deposit and they instead regulate an economy of a country (not sure about "many", but at least National Bank of Serbia is one of those).

And Bearings Bank was founded in 1762. And made it to 1995 before imploding. Likely to survive, yes. But at least one is likely to fail over any reasonable period.

In 1941 when the Nazi government stripped German Jews of their citizenship, the Swiss authorities applied the law to German Jews living in Switzerland by declaring them stateless; when in February 1945 Swiss authorities blocked German Bank accounts held in Switzerland they declared that the German Jews were no longer stateless, but were once again German and blocked their Swiss bank accounts as well.

https://en.m.wikipedia.org/wiki/Bergier_commission