<p>How and why base vibration can stabilize an inverted pendulum has puzzled the scientific community for decades, until the work on dynamic stabilization by Pyotr Kapitza pointed at the alternation between unstable and stable modes as a pathway to stability. We report the discovery of a mechanical oscillator that switches between two unstable modes, has an unstable average, and, yet, can be dynamically stabilized. Our system is governed by a modified Meissner’s model – a one-degree-of-freedom oscillator where both stiffness and damping are modulated through a square wave to switch between positive and negative values. We theoretically prove the existence of compact antiresonance windows and provide experimental evidence through a cantilever beam oscillator subject to magnetic and aerodynamic forcing. The prospect of dynamic stabilization in the absence of any stable feature has vast implications from network dynamical systems, to structural mechanics and robotics.</p>