As more experienced programmers pointed out (thank you to afoken!!!), I've had the buffering issue. After reading this article at perl.plover.com/FAQs/Buffering.html, I've applied the filehandle HOT. Now my script works as expected and looks like this:

#!/usr/bin/perl -w
use strict;
use warnings;

print "Please, enter the hex number : \n";
S|=1;

my $hex = <STDIN>;
print "$hex", "\n";

chomp ($hex);

print "You've entered hex: ", $hex, "\n";
print "The decimal equivalent is ", hex "$hex", "\n";
[download]

Thank you very much for the help!