You're most of the way there; just define your variable outside the loop and then add the lines to it. Check out the lines I added to your code below. You'll want to use one method or the other, depending on whether you want the lines in an array or as a single string in a scalar.

open(F, "file"); 
my $lines = '';  # a scalar to store the lines in
my @lines;       # an array to store the lines in
while(<F>) {
   if (/StartPattern/.../StopPattern/) {
      $lines .= $_;    # append the line to my scalar
      push @lines, $_; # add the line to my array
   }
}
close(F);
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