The increasing demands of location-based services have spurred the rapid development of indoor positioning system and indoor localization system interchangeably (IPSs). However, the performance of IPSs suffers from noisy measurements. In this paper, two kinds of robust extreme learning machines (RELMs), corresponding to the close-to-mean constraint, and the small residual constraint, have been proposed to address the issue of noisy measurements in IPSs. Based on whether the feature mapping in extreme learning machine is explicit, we respectively provide random-hidden-nodes and kernelized formulations of RELMs by second order cone programming. Furthermore, the computation of the covariance in feature space is discussed. Simulations and real-world indoor localization experiments are extensively carried out and the results demonstrate that the proposed algorithms can not only improve the accuracy and repeatability, but also reduce the deviation and worst case error of IPSs compared with other baseline algorithms.