At the beginning of the episode on the Battleship Wisconsin just before Operation Mainbrace, the senior officer mentions the Doomsday Clock and how they just advanced it to '2 minutes to midnight'. The Doomsday Clock was advanced September 1953 whereas Exercise Mainbrace was held September 1952. During the Operation, the clock was actually 3 minutes to midnight.
At the start of the episode, two officers are talking on the stern of the USS Wisconsin. They notice an Unidentified Submerged Object (USO). When the USO moves towards the battleship, you hear the command: "Forward batteries open fire. Forward batteries engage - engage". The forward batteries would be used for a threat attacking towards the bow of the ship and would be ineffective in an attack from the stern. The correct command would have been "Aft batteries open fire. Aft batteries engage - engage."
When Dr Hynek and Captain Quinn first arrive on the USS Wisconsin, they are greeted by a Chief Petty Officer (Storekeeper rate - SKC). When they are presented to the Admiral on the bridge, the Admiral addresses the sailor as "Master Chief". The events in this episode are supposed to occur during Operation Mainbrace in 1952. The paygrade of Master Chief Petty Officer was not created until 1958.
The USS Wisconsin is shown at sea, however, at the time all US Navy vessels has a large Hull Number painted on the bow. USS Wisconsin is BB 64, and the large 64 is missing from the hull.
No lights would be on in the bridge of a ship operating at night - it would make it difficult to see out the windows. Any active lights would be red.
When Dr. Hynek travels to Tierra del Fuego in the background there is a sign that says "Tierra del Fuego Chile". Tierra del Fuego is an Argentinian State, not Chilean.