This work presents an effective and robust methodology to achieve the joint estimation of direction-of-arrival (DOA) and range. One or more near-field sources are the signals of interest. Far-field sources and alpha-stable noise are additive interferences and will be removed in two steps. First, alpha-stable noise is suppressed into Gaussian or sub-Gaussian noise through a proposed mathematical limiter. Then, the near-field sources are separated from the mixed sources according to the specific technique. The two steps involve three key technologies, including noise reduction, sparse reconstruction, and subspace decomposition. In the experiments, we design an assessment standard for missing detection in data calculation, which is of utmost importance to demonstrate the novel methodology's superiority compared to the other existing competitive ones.