Taking a Constitutional

Sorry, I should have included the rest of the statement.

From the Globe:

Dry Dock 1 is the second-oldest dry dock in operation in the country, and the Constitution was the first warship to enter it, on June 24, 1833.

It looks like the oldest is in Norfolk, VA.
From Wikipedia:

Drydock Number One, Norfolk Naval Shipyard, also known as Drydock No. 1, is the site where the USS Merrimack was modified to be the Confederate Navy ironclad CSS Virginia. It is now included within the Norfolk Naval Shipyard. It remains in use.

This dry dock and another like it in Norfolk, VA were built around the same time. Loammi Baldwin, Jr. gets a lot of credit for figuring out some of the finer details with elegant solutions for this and its sibling that are still employed in modern systems. The one in Norfolk was used for repairs to the Delaware before it was fully completed in 1834, but both were first used in June, 1833.

Hard to tell which is "oldest" in that sense. Both have been enlarged, both are still in use!

The oldest working drydock is at Portsmouth, UK. It was built in 1495.