When the Constitution was written, we had privately-owned cannons.
The Constitution covers granting "Letters of Marque and Reprisal", granting permission for privately-owned warships to go after the shipping of the enemy. Private warships, with cannons.
Protected, as I said, by the second amendment. I suspect, however, that an aircraft carrier (even without the air wing, much less the battle group) is more expensive in real terms than any American privately owned armed vessel was in the late 18th/early 19th Centuries. Probably more expensive than a Ship of the Line was.