Your self-hosted blog would not be on WordPress.org. It would be on your own domain. You would have to know how to install WordPress on your own server space, or have a hosting company that offers it pre-installed.
If you're looking for requirements your hosting company must meet in order for your WP install to be successful, they're listed on the WP site.
If this is too complicated, you can have a hosted WP blog at WordPress.com.