There are a few things I'd do differently, first the open I'd write as

open my $FH, '<', $file or die "Can't open $file: $!\n";
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• The lexical file handle $FH is limited is scope to just the block of code it appears in so won't clobber file handles opened else where. Probably not a concern for a short script but it still a good habit.
• The three argument form of open where the open mode is specified separately is safer especially when $file comes from an untrusted source.
• I use low precedence or so I can get rid of the brackets
• I put $! in the error message so I know why the open failed.

The loop I'd rewite as this

use Data::Dumper;

my @arr;
while (<DATA>) {
    chomp;
    push @arr, $1 if m{^ +\s+(//.*)};
}

print Dumper \@arr;

__DATA__
  // humpty dumpty
  // sat on a wall
-// eating her curds and whey
  // humpty dumpty
  // had a great fall
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which produces

$VAR1 = [
          '// humpty dumpty',
          '// sat on a wall',
          '// humpty dumpty',
          '// had a great fall'
        ];
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Just a couple of points

• I use m{ } to avoid the leaning toothpicks.
• I use capturing parentheses to grab the bit of string I want, if there is a match it is in $1 and it is pushed onto the array