The barrier height for the most asymmetric fission decay of doubly charged sodium clusters (Na_N^2+) into singly ionized fragments has been computed with use of density-functional theory and the jellium model. We have found that the barrier is sizable for large or intermediate-size clusters, but vanishes for N<=9. We have also computed the energy DeltaH_e, needed to evaporate a neutral monomer from Na_N ^2+. For N<=40, the barrier height is smaller than DeltaH_e and emission of a Na^+ ion is the preferred decay channel of hot Na_N^2+ clusters. On the other hand, the barrier height is larger than DeltaH_e, for N>40 and, in this case, monomer evaporation becomes competitive. The critical cluster size, N_c=40, for the transition from one decay mode to the other is in reasonable agreement with the experimental result. Our calculations suggest that the mechanism for neutralmonomer evaporation is different from the one currently assumed.