Tie::File is simplest.

tie my @array, 'Tie::File', $file_name
   or die("Unable to open file \"$file_name\": $!\n");

print($array[$_])
   for 6, 1003, 2965;
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Tie::File can use a fair bit of memory (for its index, which is above and beyond what the memory argument limits), so you might want to write your own (faster, more memory efficient) solution.

my %lines_of_interest = map { $_ => 1 } 6, 1003, 2965;

open my $fh, '<', $file_name
   or die("Unable to open file \"$file_name\": $!\n");

my $num_lines = keys %lines_of_interest;

while (<$fh>) {
   if ($lines_of_interest{$.}) {
      print;
      last unless --$num_lines;
   }   
}
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Update: Added Tie::File code.
Update: Added key counting to exit loop sooner.