Angular dependent x-ray absorption fine structure ͑XAFS͒ measurements of all La, Ba, and Cu sites determined the origin and nature of the intrinsic disorder present in the low temperature tetragonal ͑LTT͒ structural ground state of La 2Ϫx Ba x CuO 4 with xϭ0.125,0.15 at Tϭ10 K. Ba doping induces major local distortions that extend as far as 5 Å from its lattice site. Despite the low Ba content the large distortions significantly affect the XAFS of the majority La and Cu sites. The distribution of the local LTT tilt angle of CuO 6 octahedra was directly determined from single and multiple scattering contributions to the XAFS of La and Cu sites. This angle, which regulates the strength of electron-lattice and spin-spin coupling in this material, was found to exhibit large, random fluctuations with rms ͓͗(Ϫ͗͘) 2 ͔͘ 1/2 Ϸ2͗͘, ͗͘ being the average tilt angle determined by diffraction techniques. These fluctuations originate in the large, random, static disorder introduced with Ba doping. Their presence has implications for the plausibility of a mobility gap causing T c suppression at xϭ0.125 in only the LTT phase, and for the correlation length of the postulated charge and spin stripes.