Nonattracting chaotic sets play a fundamental role in typical dynamical systems. They occur, for example, in the form of chaotic transient sets and fractal basin boundaries. The subject of this paper is the dimensions of these sets and of their stable and unstable manifolds. Numerical experiments are performed to determine these dimensions. The results are consistent with a conjectured formulae expressing the dimensions in terms of Lyapunov exponents and the transient life-time associated with the strange saddle.