Simulation of multivariate cross-correlated random field samples (RFSs) is often required in reliability analysis of engineering structures. Conventional parametric methods for cross-correlated RFSs simulation generally require extensive measurements to obtain reliable random field parameters (e.g., type of auto-correlation function, correlation length, and cross-correlation matrix), for characterizing both the auto-correlation and cross-correlation structures among various cross-correlated engineering quantities. However, measurement data available in practice is often limited due to time, budget, technical and/or access constraints. Therefore, it is difficult to provide an accurate estimation of random field parameters (e.g., auto-correlation and cross-correlation matrix), rendering a challenging question of how to properly simulate multivariate cross-correlated RFSs from sparse measurements, especially when the number of engineering quantities of interest is large. This study aims to address this difficulty by developing a novel cross-correlated random field generator based on Bayesian compressive sensing (BCS) and Markov chain Monte Carlo (MCMC) simulation. The proposed method is data-driven and non-parametric, and it directly uses sparse measurements as input and provides cross-correlated RFSs as output. More importantly, the proposed method is able to deal with a large number of cross-correlated quantities for big data analytics in a high-dimension domain.