One thing you might find handy is

use warnings;
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It points out potential problems with your code. (This code doesn't have any, but it is a good technique to use anyway).

In this case I think your problem is that you're double-chomping $word. When reading input you only need to chomp once to remove the line terminator (CR/LF).

print "Search for what: ";
chomp(my $search = <STDIN>);
@words = split(/ /, $search);
open(DATA, "data.txt") or die "error opening file $!";
@data = <DATA>;
foreach $data(@data) {
  foreach $word(@words) {
    if ($data =~ /$word/) {
      print $data;
    }
  }
}
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