We present X-ray and radio observations of what may be the closest type Iax supernova (SN) to date, SN 2014dt (𝑑 = 12.3 − 19.3 Mpc), and provide tight constraints on the radio and X-ray emission. We infer a specific radio luminosity 𝐿 𝑅 < (1.0 − 2.4) × 10 25 erg s −1 Hz −1 at a frequency of 7.5 GHz and a X-ray luminosity 𝐿 𝑋 < 1.4 × 10 38 erg s −1 (0.3-10 keV) at ∼ 38 − 48 days post-explosion. We interpret these limits in the context of Inverse Compton (IC) emission and synchrotron emission from a population of electrons accelerated at the forward shock of the explosion in a power-law distribution 𝑁 𝑒 (𝛾 𝑒 ) ∝ 𝛾 − 𝑝 𝑒 with 𝑝 = 3. Our analysis constrains the progenitor system mass-loss rate to be 𝑀 < 5.0 × 10 −6 M yr −1 at distances 𝑟 10 16 cm for an assumed wind velocity 𝑣 𝑤 = 100 km s −1 , and a fraction of post-shock energy into magnetic fields and relativistic electrons of 𝜖 𝐵 = 0.01 and 𝜖 𝑒 = 0.1, respectively. This result rules out some of the parameter space of symbiotic giant star companions, and it is consistent with the low mass-loss rates expected from He-star companions. Our calculations also show that the improved sensitivity of the next generation Very Large Array (ngVLA) is needed to probe the very low-density media characteristic of He stars that are the leading model for binary stellar companions of white dwarfs giving origin to type Iax SNe.