Parts of the Cloaca Maxima are still in use as a (tiny, tiny, tiny) segment of Rome's sewer system. A few aqueducts, such as the aqua Virgo, are operational as well - but they only feed monumental fountains for the water they carry is untreated. The Pantheon's been a Catholic basilica since the 7th century AD but the building's far older than that...

...and then there are the two bridges - Cestius and Fabricius - connecting the Tiberine island with the rest of the city.

I am sure there are plenty of such structures, one of which is the Tower of Hercules in northwestern Spain. Built by the Romans around the 2nd century AD, it still serves its original purpose as a lighthouse; however, it was partially restored in the 18th century.

I think the BBC actually mentioned the Tower of Hercules in an article the other day, but I can't find it again.

Not as old as you ask for, but Taos Pueblo has been more or less continuously inhabited in their present forms since at least the 18th century, and we know the location has been inhabited since the early 14th century. Whether or not the actual structures date before the early 18th century is up for debate, but seems likely. Likewise, the structures at Acoma Pueblo in New Mexico have been in use for at least 370 years and some may have been used much longer (again, perhaps up to the early 14th century).

Of course, these don't have the same antiquity as what you are looking for, but they are some of, if not the oldest, continually inhabited structures north of Mexico.

Sorry, we don't allow throughout history questions. These tend to produce threads which are collections of trivia, not the in-depth discussions about a particular topic we're looking for. If you have a specific question about a historical event or period or person, please feel free to re-compose your question and submit it again. Alternatively, questions of this type can be directed to more appropriate subreddits, such as /r/history or /r/askhistory.