I wouldn't even use perl for this, but if you must:

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

my ($string1, $string2);

# read files in chunks instead of lines
# remove if you realy want to read textfiles
# line by line
local $/ = \32768; # "block size"

open (FILE1, "< x") || die "Cannot open file : $!";
open (FILE2, "< y") || die "Cannot open file : $!";
open (FILE3, "> z") || die "Cannot open file : $!";

# set binmode, so binary files don't get messed
# up on systems with text/binary mode files
# (like windows or MS-DOS)
binmode (FILE1);
binmode (FILE2);
binmode (FILE3);

print FILE3 <FILE1>,<FILE2> or die "Error concatenating: $!";

close FILE1 or die "Error closing x: $!";
close FILE2 or die "Error closing y: $!";
close FILE3 or die "Error closing z: $!";
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update: fixed copy/paste error

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

my ($string1, $string2);

$/ = undef;

open (FILE1, "< x") || die "Cannot open file : $!";
open (FILE2, "< y") || die "Cannot open file : $!";
open (FILE3, "> z") || die "Cannot open file : $!";

$string1 = <FILE1>;
$string2 = <FILE2>;

print FILE3 join('', $string1, $string2);

close (FILE1);
close (FILE2);
close (FILE3);
[download]

Updated your answer; does it look ok now? Please /msg QandAEditors if you need changes made.

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