isn't there a danger $self->{'JOKE'} is already in use?

Indeed. One way of avoiding this is to use your own package name in the hash key, to ensure a little more uniqueness:

my $package = __PACKAGE__ . ':';

sub set_joke {
  my $self = shift;
  $self->{$package.'JOKE'} = shift;
}

sub get_joke {
  my $self = shift;
  return $self->{$package.'JOKE'};
}

sub say_joke {
  my $self = shift;
  print $self->get_joke, "\n";
}
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This technique can only get you so far though - with some classes it's not enough, so the inside-out object technique explained by JavaFan is necessary.

As an example, imagine an EmailHeaders class where each hash key represents an e-mail header (To, From, Subject, etc). You want to subclass it and store some additional information which is not an e-mail header. But any key you add to the hash will be interpreted as another e-mail header by the parent class. Here you really would need to use inside-out objects.