We propose and implement a protocol for a scalable, cost-effective, information-theoretically secure key distribution and management system. The system, called Distributed Symmetric Key Establishment (DSKE), relies on pre-shared random numbers between DSKE clients and a group of Security Hubs. Any group of DSKE clients can use the DSKE protocol to distill from the pre-shared numbers a secret key. The clients are protected from Security Hub compromise via a secret sharing scheme that allows the creation of the final key without the need to trust individual Security Hubs. Precisely, if the number of compromised Security Hubs does not exceed a certain threshold, confidentiality is guaranteed to DSKE clients and, at the same time, robustness against denial-of-service (DoS) attacks. The DSKE system can be used for quantum-secure communication, can be easily integrated into existing network infrastructures, and can support arbitrary groups of communication parties that have access to a key. We discuss the high-level protocol, analyze its security, including its robustness against disruption. A proof-of-principle demonstration of secure communication between two distant clients with a DSKE-based VPN using Security Hubs on Amazon Web Server (AWS) nodes thousands of kilometres away from them was performed, demonstrating the feasibility of DSKE-enabled secret sharing one-time-pad encryption with a data rate above 50 Mbit/s and a latency below 70 ms.