This is probably your best bet. Make a Scrolled Pane to hold the Listbox, and make a bottom frame set to -fill=>'x', -expand =>0, and the buttons are too. It's not perfect, if you make it too small, everything gets crushed.

#!/usr/bin/perl
use warnings;
use Tk;
use Tk::Pane;

my @output = (1..100);

my $log = MainWindow->new( title => "Who is logged on which tester" );
$log->geometry('125x150');
# first pack a scrolled Pane then a bottom frame
my $spane = $log->Scrolled('Pane',
           -scrollbars => 'osoe',
           -bg => 'blue',
           )->pack(-expand=>1,-fill=>'both');

#let the scrolled Pane handle the listbox scrolling
$listbox = $spane->Listbox(
   "-font"       => 14,
   "-width"      => 51,
   "-height"     => 27,
   "-background" => 'lightgreen'
)->pack(-expand => 1, -fill => 'both');

foreach $n ( @output ) { $listbox->insert( 'end', "$n" ); }

my $logU = $log->Frame->pack( -fill => 'x', -expand => 0 );

$logU->Button(
   -text             => 'Refresh',
   -background       => 'lightgreen',
   -activebackground => '#0000ff',
   -activeforeground => '#80ffff',
   -command          => sub { $Refr = 1; $log->destroy; }
)->pack( -side => 'left', qw/-ipadx 4 -pady 2/, -expand => 0 );

$logU->Button(
   -text             => 'OK',
   -background       => 'lightgreen',
   -activebackground => '#0000ff',
   -activeforeground => '#80ffff',
   -command          => \&exit
)->pack( -side => 'right', qw/-ipadx 14 -pady 2/, -expand => 0 );

MainLoop;
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