You could use a counter, which is set to 1 when you reach START then reset to 0 when you reach END. As the counter will only be 1 between START and END, you can simply print out any lines where this is the case.

use strict;

my $count = 0;
open (IN, "input.txt");
while (<IN>) {
    if (/START/) {
    $count = 1;
    }
    elsif (/END/) {
    $count = 0;
    }
    elsif ($count) {
    print;
    }
}
close IN;
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Output:

These are the first

set of lines

which are to be extracted


These are the second

set of lines

which are to be extracted
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I wasn't sure if the blank lines were intentional, but if you don't want them, change to

elsif ($count && $_ !~ /^$/)
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