You attempt to read from $data, yet never open it that file handle.
You changed the initial content of @data. (It should be empty.)
DATA shouldn't be used as an all-purpose file handle. It's a special file handle.

Fix:

#!/usr/bin/perl -w

use strict;

my $data_qn = 'data_file';

open($data_fh, '<', $data_qn)
   or die("Unable to open plot data \"$data_qn\": $!\n");

my @data;
while (<$data_fh>) {
   my @fields = split ' ';
   for my $row (0..$#fields) {
      push @{$data[$row]}, $fields[$row];
   }
}

use Data::Dumper;
print Dumper \@data;
[download]

Update: Oops, forgot to re-add the my @data; that was removed. Fixed.