Stardate
A stardate is the official reference to the date in the Star Trek universe instead of referring to our Gregorian calender. A stardate consists of four decimal numbers and a single decimal place.
During the making of the various Star Trek series and movies, no clear algorithms for converting the stardates to the Gregorian calender were used and therefore, today, a lot of controversy exists around the conversion of the dates, in particular in between the different series.
For further information about the different theories on stardates see Wikipedia and the TrekGuide website.
In The Nerdvana Annihilation, Sheldon converts stardate 5027.3 to January 10, 2328, a result that will also be obtained using the online Stardate converter.
However, the episode he is referring to, "The Enterprise Incident" from the third season of Star Trek: The Original Series, is officially dated to take place in 2268 our time [1].
In The Adhesive Duck Deficiency, Sheldon records his personal log and states that it is "Stardate 63345.3". This converts to November, 17th, 2009, 3pm (GMT, which would be 7 am in Pasadena), according to Trekguide's stardate converter. This day coincides with the peak of the Leonid meteor shower and thus places this episode on Tuesday, 17th November, 2009 in our timeline.
However, the time of the day seems to be slightly incorrect, since it is dark outside the window of the living room when Sheldon records his log and it is also dark in the desert where the guys have their camp. Leonard states it would be 5 hours until the peak of the meteors, suggesting the episode starts on Tuesday evening rather than morning. We should assume this slight inaccuracy stems from the variations in the stardate converters, see discussion above.