We report on the fabrication of a ballistic SNS junction with a two-dimensional electron gas (2DEG) in an InGaAs-based heterostructure. We have observed the Josephson current as well as the subharmonic energy-gap structures caused by the MAR. Moreover, we experimentally estimated the thermal noise by comparing measured IV characteristics with those obtained with an extension of the Ambegaokar and Halperin theory. As a consequence, we have observed giant thermal noise that is much larger than that expected with normal reservoirs. The experimentally obtained "giant" thermal noise can be qualitatively explained by the Martín-Rodero theory that considers both the ballistic transport of the 2DEG and the thermal fluctuation in the coherent multiple Andreev reflection regime.