What is the closest star to Earth other than the sun?

a=Most@Sort[StarData[
  EntityClass["Star", "StarNearest10"], {"Name", 
   "DistanceFromSun"}], #1[[2]] < #2[[2]] &]

(*{{"Proxima Centauri",   Quantity[4.2181,  "LightYears"]}, 
   {"Rigel Kentaurus A",  Quantity[4.38982, "LightYears"]}, 
   {"Rigel Kentaurus B",  Quantity[4.4001,  "LightYears"]}, 
   {"Barnard's Star",     Quantity[5.9339,  "LightYears"]}, 
   {"Wolf 359",           Quantity[7.78813, "LightYears"]}, 
   {"Lalande 21185",      Quantity[8.30217, "LightYears"]}, 
   {"Luyten 726-8 B",     Quantity[8.5573,  "LightYears"]}, 
   {"Luyten 726-8 A",     Quantity[8.5573,  "LightYears"]}, 
   {"Sirius",             Quantity[8.59093, "LightYears"]}*)

TimelinePlot[
 Association[(a[[#, 1]] -> a[[#, 2]]) & /@ Range@Length@a], 
 DateFunction -> (DatePlus[DateObject[Round[#*365*24*60*60, 1]], 
     Quantity[-1900, "year"]] &), FrameLabel -> "Light Years", 
 PlotLabel -> "Distance from Sun"]

To clarify why it is that your command did not work, here is what your free-form input was translated to:

EntityValue[EntityClass["Star", {"DistanceFromSun" -> TakeSmallest[1]}], "DistanceFromSun"]

This returns {Missing["NotAvailable"]} because the Sun itself is part of the "Star" domain and though it is closest to itself it (apparently) has no value for "DistanceFromSun". Note that this appears to be a bad job on the part of the free-form interpreter... the command closer to what I imagine you are interested in would be this:

EntityValue[EntityClass["Star", {"DistanceFromSun" -> TakeSmallest[1]}], "Name"]

in which case we get the output of {Sun} (which, though it is not the answer you are looking for, at least the problem would have been clearer).

Now to solve this problem, use TakeSmallest[2] instead of TakeSmallest[1] and look for the property "Name" instead of "DistanceFromSun" in EntityValue:

EntityValue[EntityClass["Star", {"DistanceFromSun" -> TakeSmallest[2]}], "Name"]

in which case we get the expected output of {Sun, Proxima Centauri}. I imagine though that you would probably only want the second element of this list, so you could do

EntityValue[EntityClass["Star", {"DistanceFromSun" -> TakeSmallest[2]}], "Name"][[2]]

and get the single output Proxima Centauri, as desired.

If you also want the distance to Proxima Centauri, you can do the following:

EntityValue[EntityClass["Star", {"DistanceFromSun" -> TakeSmallest[2]}], {"Name", "DistanceFromSun"}]

and you will get this output, as expected:

{{Sun, Missing["NotAvailable"]}, {Proxima Centauri, 4.2181 ly}}

Note that, as I mentioned, the "DistanceFromSun" for the Sun itself is Missing["NotAvailable"] and this is the output you originally received.