Root SSH access is not given in shared hosting, for many reasons. No shared hosting environment allows root access to their shared hosting server. This would compromise customers data, and would cause lots of issues. You can however, get a limited SSH account limited to your home directory, which can do tasks that you need to do. You cannot edit server configuration, access sensitive directories, stop/start services, execute programs, reboot the server, etc. You can use mysqldump to dump your MySQL databases, you can configure cron jobs in cpanel, and many other things that you need. Ultimately, if total access is needed (root) you would need to buy a VPS (Virtual private server) or a dedicated machine depending on your resource usage. If you need something installed, like a php package or something on the server would give give you trouble, support is there to assist you.
Excellent response, really nothing much to add here. On a VPS you have unrestricted SSH access. On our shared hosting we use CageFS.
CageFS is a virtualized file system and a set of tools to contain each user in its own 'cage'. Each customer will have its own fully functional CageFS, with all the system files, tools, etc...
The benefits of CageFS are:
•Only safe binaries are available to user
•User will not see any other users, and would have no way to detect presence of other users & their user names on the server
•User will not be able to see server configuration files, such as Apache config files.
•User's will have limited view of /proc file system, and will not be able to see other' users processes