Final answer:
Role-based access control (RBAC) is the system that enables only backup administrators to access all servers on the network, as it assigns permissions based on organizational roles.
Step-by-step explanation:
The access control that allows only backup administrators access to all servers on the network is Role-based access control (RBAC). In RBAC, access permissions are based on the roles within an organization, and only those individuals assigned to specific roles are granted access to resources. This ensures that backup administrators, who are assigned a specific role that necessitates access to all servers for performance of backups, are the only ones with such comprehensive access.