We have investigated the low-temperature physical properties of BaTi₂Sb₂O andBa_1−xNa_xTi₂Sb₂O (x = 0.05, 0.1, 0.15, 0.2, 0.25, 0.3) by means of muon spin rotation (μSR) and SQUID magnetometry. Our measurements reveal the absence of magnetic ordering below T_DW = 58Kin the parent compound. Therefore the phase transition at this temperature observed by magnetometry is most likely due to the formation of a charge-denisty wave (CDW). Upon substitution of barium by sodium in Ba_1−xNa_xTi₂Sb₂O we find for x = 0.25 superconductivity with a maximum T_c = 5.1 K in the magnetization and a bulk T_c,bulk = 4.5 K in the μSR measurements. The temperature dependency of the London penetration depth λ⁻²(T) of the optimally doped compound can be well explained within a conventional weak-coupling scenario in the clean limit.