E.g., a church that is still used as a church, or a home that is still used as a home.
Edit: As a follow up, what would the answer be if religious/ceremonial sites were excluded?
Perhaps something like Epidaurus Theatre. It was built in 4th century BC as a theatre, it is still used as a theatre, though it wasn't used continuously all that time. I'm not sure if an even older theatre exists that is still used.
If you're excluding graves/tombs, then the Pantheon in Rome seems to fit the bill
Possibly Etchmiadzin Cathedral in Armenia, from ~301 CE. But there are a bunch of extant churches from this period still around, including St Peter's.
If you discount religious architecture, as well as extant bridges, dams and walls, one of the oldest continually used buildings in the UK would be Westminster Hall from 1097. Windsor Castle has been a royal residence since the 1100s. I'm curious what the other contenders are.
Among the non-religious category, are we counting bridges? If so, there are claims that a bridge in Turkey, which is still in use, dates back to 850 BC.
There are multiple sites reporting this, but I could not find any respectable sources. Maybe someone with suitable expertise can weigh in?
This is the Caravan Bridge over the river Meles in Izmir, Turkey. It was built around 850 BC, which makes it more than 2,860 years old—qualifying as the oldest functioning bridge in the world.
Good question. The Pantheon in Rome been a temple since around 100AD or so, there is another Roman temple in France (the name escapes me) that may be a bit older, but if I recall it has had a lot of work done to it, whereas the Pantheon hasn't.