My favorite thing about this video, other than the baby with the mustached guys, is that, though the band could sure enough have shot this video in Clarksville, which is right up the road from Nashville, they clearly filmed it in Nashville and out at the Natchez Trace. By the time you get from downtown out to the Trace, you could be from downtown almost to Clarksville.
Tiny Cat Pants 
There never was a train to Clarksville. The way I heard the story is that the song was written or recorded in Nashville and they just took out a map and picked a name that sounded right.
No, there has to have been a train to Clarksville. Every town on a train track got passenger trains until the rise of the automobile. Plus, with the base being right there, they were moving those soldiers via rail until moving them via plane became more cost effective.
The trick is to figure out if there was passenger service to Clarksville still when that song was written.
But man, when you think about the base being there, it makes “Last Train to Clarksville” one of the saddest upbeat songs ever–“Come to Clarksville before I ship out to Vietnam and possibly die and we will kiss and drink coffee.”
Ooo, the Montgomery County Historical Society answers the question!
http://www.mchsociety.org/Sub-Pages/Hist-TrainStation.html
Though, warning, the song plays automatically and loudly.