They can refuel ships at sea so I assume they can keep the jet fuel topped up. They don't have any ability to reload ship-launched missiles at sea, though, so if they run out they have to go back to a port with the capability to resupply those. I understand they fire two anti-missile missiles for every interception attempt, so it seems pretty likely they'd have ripped through their magazines pretty quickly.

Yes. Shit tons of fuel. Stored around and underneath on the bottom of the carriers. We used to grab a bit when we needed to clean up paint spills. JP5 is some potent stuff.

Then they duck out and get regular refuelings at sea. These refuelings at sea are a particualr vulnerable time for both the merchant marine tanker ship and the carrier. They are very very close to each other, hooked up by multiple lines, have almost no maneuverability during the procedure.