The performance of near-surface disposal facilities (NSDFs) for low-level radioactive waste depends on the type of materials used in the construction of barrier system, the properties of geological formations around the facility, and the type of waste containment system. By taking into account the effect of the factors mentioned here, a mathematical model can be developed to evaluate the migration of contaminants and radionuclides (via the contaminant transport model) from NSDFs to the nearest geosphere and biosphere, and assess the performance of the whole system. The model for migration of contaminants in a multibarrier system is considered in this paper for the safety assessment of NSDFs. The model considers scenarios during dumping period, after the dumping period, and after the closure of the NSDF. The annual radiation dose due to a radionuclide through drinking water pathway (for the single dump mode and multiple dump mode) is evaluated with the model. In the present work, a comprehensive analysis is carried out to estimate the radiation dose of seven radionuclides (, , , , , , and ) for different modes of disposal. Among these radionuclides, radioactive carbon () delivered the maximum concentration in groundwater. To consider variability in the model parameters and quantify the uncertainties due to inherent variability, the presence of a heterogeneous medium, and variability associated with the long time scales of interest, a probabilistic method called the collocation-based stochastic response surface method (CSRSM) is used. In this method, a complex analytical equation is approximated by a higher-order polynomial [using polynomial chaos expansion (PCE)]. Radioactive carbon () is used for probabilistic analysis, considering groundwater velocity, thickness of unsaturated zone, dispersivity, and distribution coefficient as random variables. In all the scenarios, a third-order polynomial gave the best fit for the model and coefficient of determination () of 0.99 was obtained for a third-order polynomial. Reliability analysis was carried out, and the probability of failure of an annual radiation dose of exceeding the permissible limits was estimated for different scenarios. The results show that the probability of failure of system in all the scenarios is very low, confirming the adequacy of the system.