This doesn't work:

my $obj = WordNet::Similarity::"$Measure"->new($wn);

A better solution is to put the entire package name in a variable:

my $measure_class = "WordNet::Similarity::whatever";
my $measure = $measure_class->new();

...
sub similarity {
    ...
    return $measure->getRelatedness($w1, $w2);
}
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Besides fixing the syntax error, this will also create the measure object only once at the beginning of your script.

Also, note that using @words = <FILE> will leave the end of line characters on your strings, but maybe that's not a problem. If it is, just chomp your words:

chomp(@words = <FILE>);