We report on zero-field muon spin rotation, electron-spin resonance, and polarized Raman scattering measurements of the coupled quantum spin ladder Ba2CuTeO6. Zero-field muon spin rotation and electron-spin resonance probes disclose a successive crossover from a paramagnetic through a spin-liquid-like into a magnetically ordered state with decreasing temperature. More significantly, the two-magnon Raman response obeys a T-linear scaling relation in its peak energy, linewidth, and intensity. This critical scaling behavior presents an experimental signature of proximity to a quantum-critical point from an ordered side in Ba2CuTeO6.