The triple perovskite oxide Ba₃MnTa₂O₉ has been synthesized, and its magnetic properties have been investigated through dc and ac magnetization, specific heat, electron spin resonance measurements, and density functional theory (DFT) calculations. Mn²⁺ (S = 5/2) ions are the only magnetic species present in the material. These Mn²⁺ ions constitute a quasi-two-dimensional triangular network in the crystallographic ab plane. Magnetization and specific heat measurements reveal the absence of any long-range magnetic order down to 0.5 K despite the presence of antiferromagnetic correlations between the magnetic ions, suggesting the presence of geometric frustration in the material. The entropy release is lower than the expected theoretical value of R ln(6), further suggesting the presence of frustration. First-principles calculations using DFT and atomistic spin dynamics simulations further support this lack of static magnetic order even at low temperatures and identify the competing magnetic interactions along with the quasi-two-dimensional magnetic dimensionality as the underlying origin of this unconventional magnetic behavior.