The work presented in this paper is about implementing a frequency-modulated continuous wave (FMCW) multiple-input multiple-output (MIMO) positioning radar and a sparse spectrum fitting (SpSF) algorithm for range and angular measurements. First, we designed a coherent FMCW MIMO radar system working in the S-band with low power consumption that consists of four transmitter and four receiver antennas and has the ability to extend its virtual aperture; thus, this system can achieve a higher resolution than conventional phased array radars. Then, the SpSF algorithm was designed for estimating the distance and angle of the targets in the FMCW MIMO radar. Due to the fact that the SpSF algorithm can exploit the spatial sparsity diversity of a signal, the SpSF algorithm that is applied in the designed MIMO radar system can achieve a better estimation performance than the multiple signal classification (MUSIC) and Capon algorithms, especially in the context of small snapshots and low signal-to-noise ratios (SNRs). The simulated and experimental results are used to prove the effectiveness of the designed MIMO radar and the superior performance of the algorithm.