I'm not sure if the date is referenced in the TAS episode. I seem to recall a date reference in Bjp Trimble's Star Trek Concordance
which set the discovery of the stasis box within the Sol system in the late 20th or early 21st century. And remember in TOS' continuity humans had been out into the solar system by the 1990s. In that continuity the space programs didn't pull back after Apollo or else DY class ships would never have been developed by the 1990s.