PS. - Wouldn't this be a great way to teach astronomy to kids...
A starship like the Enterprise would explore a different astronomical object, (a binary, variable, cluster, nebula, supernova remnant, collapsar, etc) every week. No aliens, battles, no love affairs, no plot, just a survey ship routinely cataloging and mapping our little corner of the galaxy. Every week some bizarre new object would appear on the screens and the data would come in through the ship's sensors while the crew discussed and explained what they were observing.
You might even teach a little basic science and engineering, like how airlocks work, and why their doors have to be made to only open inwards...