The Casimir interaction between two parallel metal plates in close proximity is usually thought of as an attractive interaction. By coating one object with a low-refractive index thin film, we show that the Casimir interaction between two objects of the same material can be reversed at short distances and preserved at long distances so that two objects can remain without contact at a specific distance. With such a stable Casimir equilibrium, we experimentally demonstrate passive Casimir trapping of an object in the vicinity of another at the nanometer scale, without requiring any external energy input. This stable Casimir equilibrium and quantum trapping can be used as a platform for a variety of applications such as contact-free nanomachines, ultrasensitive force sensors, and nanoscale manipulations.