Select[FileNames["*", dir, Infinity],
   Not@*DirectoryQ] // Map[FileByteCount] // Total

Select[FileNames["*", "~/Documents", Infinity], Not@*DirectoryQ] //
   Replace[
     Quiet@Map[FileByteCount, #],
     Except[_Integer] -> 0,
     1
     ] & // Total // AbsoluteTiming

{2.38783, 6161484250}

ToExpression@
     StringSplit[
       RunProcess[{"du",  "-s", ExpandFileName@#},
        "StandardOutput"]][[1]]*1024/2 &@
  "~/Documents" // AbsoluteTiming

{0.352957, 6080770048}