Watched in the 70s and watched again in the last few years on the old reruns.
Obviously the most unrealistic part is the personal greeting from the "stars of the ship" (stars of the show) as you board. On an episode I saw recently, Captain Stubing took a liking to a lady as she was boarding, started wooing and romancing her immediately, invited her for private dining, went to her cabin, etc.
Made me wonder who was in charge of navigation, personnel, etc.