At room temperature, mechanical motion driven by the quantum backaction of light has been observed only in pioneering experiments in which an optical restoring force controls the oscillator stiffness1,2. For solid-state mechanical resonators in which oscillations are controlled by the material rigidity, the observation of these effects has been hindered by low mechanical quality factors, optical cavity frequency fluctuations3, thermal intermodulation noise4,5 and photothermal instabilities. Here we overcome these challenges with a phononic-engineered membrane-in-the-middle system. By using phononic-crystal-patterned cavity mirrors, we reduce the cavity frequency noise by more than 700-fold. In this ultralow noise cavity, we insert a membrane resonator with high thermal conductance and a quality factor (Q) of 180 million, engineered using recently developed soft-clamping techniques6,7. These advances enable the operation of the system within a factor of 2.5 of the Heisenberg limit for displacement sensing8, leading to the squeezing of the probe laser by 1.09(1) dB below the vacuum fluctuations. Moreover, the long thermal decoherence time of the membrane oscillator (30 vibrational periods) enables us to prepare conditional displaced thermal states of motion with an occupation of 0.97(2) phonons using a multimode Kalman filter. Our work extends the quantum control of solid-state macroscopic oscillators to room temperature.|A room-temperature demonstration of optomechanical squeezing of light and measurement of mechanical motion approaching the Heisenberg limit using a phononic-engineered membrane-in-the-middle cavity with ultralow noise.