The main question studied in this article may be viewed as a nonlinear analogue of Dvoretzky's theorem in Banach space theory or as part of Ramsey theory in combinatorics. Given a finite metric space on n points, we seek its subspace of largest cardinality which can be embedded with a given distortion in Hilbert space. We provide nearly tight upper and lower bounds on the cardinality of this subspace in terms of n and the desired distortion. Our main theorem states that for any epsilon>0, every n point metric space contains a subset of size at least n^{1-ε} which is embeddable in Hilbert space with O(\frac{\log(1/ε)}ε) distortion. The bound on the distortion is tight up to the log(1/ε) factor. We further include a comprehensive study of various other aspects of this problem.