The following (untested) code should help:

use strict;
use warnings;

open IN, '<', "file1" or die "Failed to open file1: $|";
open IN2, '<', "file2" or die "Failed to open file2: $|";
open OUT, '>>', "joinfile" or die "Failed to open joinfile: $|";

while (! eof (IN) || ! eof (IN2)) {
    my $file1 = <IN> || '!! missing file1 line !!';
    my $file2 = <IN2> || '!! missing file2 line !!';
    chomp $file1;
    chomp $file2;
    print OUT "$file1 $file2\n";
}
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Note that a few things have been cleaned up:

• strictures have been added
• The opens have changed to 3 parameter opens
• Diagnostic error messages have been added to the dies

If you want to stop when either file runs out of lines change the || to an &&.