It is possible to run both the server and client without root privileges.
The server, by default, runs as a non-root user. However, it expects to be run as root and changes user to a specified user as soon as it can, simply for administrative convenience. The server uses a port greater than 1024, so it doesn't need root to start.
To run it entirely as a non-root user, edit the bbstored.conf file, and remove the User directive from the Server section. Then simply run the server as your desired user.
The client requires root for normal operation, since it must be able to access all files to back them up. However, it is possible to run the client as a non-root user, with certain limitations.
Follow the installation instructions, but install the executable files manually to somewhere in your home directory. Then use bbackupd-config to configure the daemon, but use a directory other than /etc/box, probably somewhere in your home directory.
All directories you specify to be backed up must be readable, and all files must be owned by the user and readable to that user.
Important: If any file or directory is not readable by this user, the backup process will skip that file or directory. Keep an eye on the logs for reports of this failure.
Non-root operation of the backup client is recommended only for testing, and should not be relied on in a production environment.