You check if the file named by the value of $datafile is writable, then you try to open the file named $datafile which doesn't exist. Get rid of those quotes!

By the way, if you declared $x and $y where you actually gave them a value, you wouldn't be trying to print out variables you haven't initialized.

#!/usr/bin/perl -w

use strict;
use warnings;

use lib "/usr/local/lib/perl/lib/perl5/site_perl/5.8.8";

use CGI::Carp qw(warningsToBrowser fatalsToBrowser);

use CGI qw(:standard);

print header;

my $datafile = "/full/path/to/file/data.txt";
system('perl', '/full/path/to/file.cgi')==0
    or die "system failed $?/$!";    

open(my $fh, '<', $datafile)
   or die "Can't open file $datafile: $!\n";

chomp( my $x = <$fh> );
chomp( my $y = <$fh> );

print "$x<br>\n";
print "$y<br>\n";
[download]