Passive magnetic bearings are an attractive alternative to conventional ball bearings in applications where wear and tear must be minimal and cost or construction restrictions ban the use of active magnetic bearings. They also find applications as miniaturized high-speed bearings where no other low-friction solutions are available. We show that, for a given cylindrical construction volume, an optimized repulsive bearing design with respect to a maximal radial stiffness can be found. This optimal configuration is independent of the aspect ratio of the bearing, but is determined by the width of the air gap between rotor and stator. Our method can determine the optimal configuration for any bearing geometry. We develop a design for a miniaturized high-speed bearing as an example.