The problem of constructing confidence regions for the median in the nonparametric measurement error model (NMEM) is considered. This problem arises in many settings, including inference about the median lifetime of a complex system arising in engineering, reliability, biomedical, and public health settings. Current methods of constructing CRs are discussed, including the T-statistic based CR and the Wilcoxon signed-rank statistic based CR, arguably the two default methods in applied work when a confidence interval about the center of a distribution is desired. Optimal equivariant CRs are developed with focus on subclasses of of the class of all distributions. Applications to a real car mileage efficiency data set and Proschan's air-conditioning data set are demonstrated. Simulation studies to compare the performances of the different CR methods were undertaken. Results of these studies indicate that the sign-statistic based CR and the optimal CR focused on symmetric distributions satisfy the confidence level requirement, though they tended to have higher contents; while two of the bootstrap-based CR procedures and one of the developed adaptive CR tended to be a tad more liberal but with smaller contents. A critical recommendation is that, under the NMEM, both the T-statistic based and Wilcoxon signed-rank statistic based confidence regions should not be used since they have degraded confidence levels and/or inflated contents.