bless \$self;
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Here, you're blessing a reference to a hash reference. You want to bless the hash reference instead:

bless $self;
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Also, you will likely want to make your class subclassable, so use the two-argument form of bless:

sub new {
  my ($class) = @_;
  my $self = {};
  $self->{data} = {};
  $self->{data}->{start_url} = "https://www.w.com/home.html";

  bless $self, $class;
}
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Also, I've removed the parentheses from the subroutine declaration, as methods ignore any declared prototypes.