There are some third-party solutions for back-up that are listed in the Joomla Extension Directory at joomla.org. You would need access to the Joomla admin to install them. But in order to move it anywhere else, you are going to need access to create a new MySQL database.
My professional recommendation is not to use this method unless you have no other choice though. In case you and others don't know, the preferred method is to create a tar ball of the website and do a mysqldump of the database. Once the files are restored along with the database, the configuration.php file that resides in the root directory of the Joomla installation needs to be edited for the new configuration.