You're getting a double callback because of this line in your code:

    $f2 -> focus;
[download]

Don't do that; it's triggering a focusout event on $f2, causing $f2 to enter its validation callback.

Note too, the first time you validate on $entryOne or $entryThree the corresponding column 2 Entry widget $entryTwo or $entryFour will not have been selectable, so the focus will skip to the other first-column Entry widget. If this isn't what you want, you could change the disabled states to readonly instead, because according to the Entry widget docs:

Switch: -state
... If the entry is readonly, then the value may not be changed using 
+widget commands and no insertion cursor will be displayed, even if th
+e input focus is in the widget; the contents of the widget may still 
+be selected. If the entry is disabled, the value may not be changed, 
+no insertion cursor will be displayed, the contents will not be selec
+table, ...
[download]

Finally, a suggestion (which you are free to take as you will). I find it useful to simplify Tk code by writing subroutines that consolidate common values. Since you're creating 4 Entry widgets with the same validation method, and your validation calls the same 2 subroutine, why not extrapolate that to the 2 new subroutines (here called entry and config_validate):

#!C:\Windows\Config\UMS_Scripts\perl\bin\perl
use strict;
use warnings;
use Tk;
my $globalVar = 0;

#################
## Subroutines ##
#################
sub first {
    my ($f1, $f2) = @_;
    $globalVar = $globalVar + 1;
    print "in sub first for the $globalVar time\n";
    my $count = 1;
    for my $anArg (@_) {
        if (defined $anArg) {
            print "$count - $anArg\n";
        } else {
            print "$count\n";
           }
        $count = $count + 1;
    }
    my $value1 = $f1 -> get();
    my $length1 = length $value1;
    print "value1: $value1 :$length1\n";
    my $value2 = $f2 -> get();
    my $length2 = length $value2;
    print "value2: $value2 :$length2\n";
    if ($length1 > 0) {
        $f2 -> configure(-state=>'normal');
        ## Don't do this -- it invoke the focusout event prematurely
        # $f2 -> focus;
        # $f2 -> update;
    } else {
    }
}


sub second {
   my ($f1, $f2) = @_;
   print "in sub second\n";
   ## You probably don't need this -- unused in this subroutine
   # $globalVar = $globalVar + 1;
   my $count = 1;
   for my $anArg (@_) {
      if (defined $anArg) {
         print "$count - $anArg\n";
      } else {
         print "$count\n";
      }
      $count = $count + 1;
   }
   my $value1 = $f1 -> get();
   my $length1 = length $value1;
   print "value1: $value1 :$length1\n";
   my $value2 = $f2 -> get();
   my $length2 = length $value2;
   print "value2: $value2 :$length2\n";
}


##################
## Main Program ##
##################
my $Mw = MainWindow->new(-title=>'Collecting Passwords');

my $entryOne   = entry($Mw, 0, 0, "normal");
my $entryTwo   = entry($Mw, 0, 1, "disabled");
my $entryThree = entry($Mw, 1, 0, "normal");
my $entryFour  = entry($Mw, 1, 1, "disabled");

config_validate($entryOne,   $entryTwo);
config_validate($entryThree, $entryFour);

MainLoop;


#####################
## New Subroutines ##
#####################
sub entry {
    my ($parent, $row, $col, $state) = @_;
    my $entry = $parent->Entry(-validate => 'focusout');
    $entry->grid(-row => $row, -column => $col);
    $entry->configure(-state => $state);
    return $entry;
}


sub config_validate {
    my ($ent_A, $ent_B) = @_;
    $ent_A->configure(-validatecommand => sub { first($ent_A, $ent_B) 
+});
    $ent_B->configure(-validatecommand => sub { second($ent_A, $ent_B)
+ });
}
[download]

Again, you're free to take it or leave it, but I find the simplification makes a Tk program a lot easier to read. Plus, you can easily see where to change the 2 occurrences of "disabled" to "readonly", to see the resulting effect.

Good luck!

Update:   Removed some parameters left over from debugging the root cause of the problem.

Update 2:   Link directly to description of State in the Tk docs for the Entry widget.