Here’s a possibility for a Cub Scout day trip from Apex. It is a little over 2 hours away in Wilmington, NC. Located on the Cape Fear River the USS North Carolina Battleship makes a great place to visit for scouts. The Battleship is open every day of the year. However, I would recommend cooler days of the Fall, Winter or Spring as there is no A/C on the ship and it is quite hot during the Summer. Our family just visited here on our recent visit to the NC Coast.
I would recommend this visit in-particular to the older scouts as there are many narrow stairs within the decks of the ship. The ship is in no way wheelchair accessible except for the top deck. Plan to wear good shoes as there is a lot of walking and of course the stairs.
Taking your time on the ship, the tour should last around 2 hours. There are a few working mechanisms on the ship that the boys can spent quite some time on. This includes the gun turrets on the port side of the ship. (Left side facing the bow of the ship).
More info can be found on the USS North Carolina Website including admission prices. Today, they are $12 for 12 and up, $6 for 6 through 11, 5 and under free.
They would like to eventually open the ship up to overnight camping but the ship is in need of restoration and repairs before that can actually happen.