In this contribution, we present a novel surrogate model, term as State-Space Kriging (S2K) [1,2], to emulate the response of high-dimensional nonlinear stochastic dynamical systems. S2K operates by approximating dynamical systems in their state-space form, where every stochastic excitation turns out to be a one-dimensional parameter. This approach effectively avoids the high-dimensionality associated with discretizing the stochastic excitation. For scenarios with partial observations of the state variables, we integrate S2K with time-delay embedding method [3]. This technique complements the non-observable spatial state variables by incorporating the time-delayed state variables at a single or multiple spatial locations. Furthermore, we apply a projection-based model reduction technique to alleviate the computational burden of the high-dimensional state-space equations in systems with large numbers of degrees of freedom and long time memory. Numerical experiments demonstrate that the proposed S2K model delivers stable and accurate long-term predictions, even when trained on only a handful of recorded time histories.