while (my $col1 = <FILE1>) {
    chomp $col1;
    last unless (my $col2 = <FILE2>);
    print "$col1\t$col2";
}
[download]

FWIW this can be written in Perl 6 very nicely:

$ perl6 -e 'for lines("a") Z lines("b") -> $a, $b { say "$a\t$b" }'
[download]

Where "a" and "b" are the names of the two files.

If the iteration should not stop after the second file is exhausted, we can pad the lines of the second file with an infinite number of empty strings:

$ perl6 -e 'for lines("a") Z lines("b"), '' xx * -> $a, $b { say "$a\t
+$b" }'
[download]

Works with Rakudo today.

Perl 6 - links to (nearly) everything that is Perl 6.

Well, that's better than what the original script does, but I'd prefer to have the end of the longer file in the output as well, paired with empty "cells".

I.e. either

bla[tab]
bla[tab]
bla[tab]
[download]

or

[tab]bla
[tab]bla
[tab]bla
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A slight mod to BrowserUk's code is what you need:

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

open ONE, '<', "one.txt" or die "First File: $!\n";
open TWO, '<', "two.txt" or die "Second File: $!\n";

until( eof( ONE ) and eof (TWO)) 
{
    my $one = <ONE>;
    my $two = <TWO>;
    $one ||= "";
    $two ||= "";
    chomp($one);
    chomp($two);
    print "$one\t$two\n";
}
__END__
OUTPUT:
a   this is
0   file two
ab  
dsf 

one.txt:
a 
0
ab  
dsf

two.txt:
this is
file two
[download]