Virtual Super-resolution Optics with Reconfigurable Swarms (VISORS) is a distributed telescope mission for high-resolution imaging of the Sun using two 6U CubeSats flying in formation in a Sun-synchronous low-Earth orbit. An optics spacecraft carries a photon sieve acting as a high-resolution lens in the extreme ultraviolet spectrum, while the image passing through the sieve is focused on a detector spacecraft. This paper presents the newly conceived design of the on-board guidance, navigation and control (GNC) system, which is highly autonomous, robust, passively safe, and validated under realistic mission simulations. The primary objective of the GNC system is to establish a passively safe and high-precision formation alignment at 40-meter separation, with sub-centimeter relative navigation and position control accuracy, over repeated observations of 10-second duration. Science mission success rates are assessed via Monte-Carlo analyses under realistically modelled uncertainties stemming from sensing errors, maneuver errors, unmodelled dynamics, and erroneous knowledge of internal spacecraft components. Precise real-time relative navigation is achieved by carrier phase differential GPS with integer ambiguity resolution. Precise control over short baselines is achieved via closed-loop optimization-based stochastic model predictive control with centimeter-level accuracy. Control at far range and during approach is achieved by closed-form impulsive control with meter-level accuracy. Passive safety is enforced throughout the mission to mitigate collision risks even under critical subsystem failure. Beyond VISORS, this work also realizes the crucial insight that the described GNC architecture is generalizable to other distributed space missions where accuracy and fault-tolerant safety are key requirements, such as rendezvous, proximity operations, and swarming missions.