The oldest warship in the U.S. Navy is currently the carrier Kitty Hawk (CV-63), which was commissioned on April 29, 1961. Its not unusual for large warships to serve for extended period. Being more sturdy, and more likely to get major overhauls, they stay in service as long as they are useful. The 86,000 ton Hawk has handled 373,157 successful aircraft landings in that time and had 32 skippers. The ship had an extensive overhaul from 1987 to 91, which makes it capable of serving into the next decade. The Hawk became the oldest navy ship in commission in 1998, and thus received the First Navy Jack (the Dont Thread on Me flag flown by the first navy warships.) The oldest ship in the navy is the only one that can fly the First Navy Jack.
The First Navy Jack moved around a lot in the 1990s, as the navy downsized because the Soviet navy had largely disappeared. The USS Prairie, the last of the pre-war US Navy ships that fought in World War II was decommissioned on 27 March 1993. The ship, a destroyer tender (a supply and maintenance ship for deployed destroyers) entered service in late 1939. The USS Prairie passed the First Navy Jack on to the USS Orion, a submarine tender commissioned in 1944. But the Orion went out of service later that year and the First Navy Jack rapidly moved from one retiring ship to another until the Kitty Hawk got it in 1998, and held on to it. The Hawk has always been considered a hard luck (not bad luck) ship, getting into interesting, or embarrassing, situations. But this ship has always been in good shape. And now its the last one of its era still in service.
Technically, the oldest ship in navy that is still in commission, is the sail powered frigate the USS Constitution. But this is a memorial and museum ship and the "commissioned" status is basically honorary. The two century old Constitution can no longer sail, or fight.