Millimeter-wave (mmWave) communication in combination with massive multiuser multiple-input multiple-output (MU-MIMO) enables high-bandwidth data transmission to multiple users in the same time-frequency resource. The strong path loss of wave propagation at such high frequencies necessitates accurate channel state information to ensure reliable data transmission. We propose a novel channel estimation algorithm called BEAmspace CHannel EStimation (BEACHES), which leverages the fact that wave propagation at mmWave frequencies is predominantly directional. BEACHES adaptively denoises the channel vectors in the beamspace domain using an adaptive shrinkage procedure that relies on Stein's unbiased risk estimator (SURE). Simulation results for line-of-sight (LoS) and non-LoS mmWave channels reveal that BEACHES performs on par with state-of-the-art channel estimation methods while requiring orders-of-magnitude lower complexity. To demonstrate the effectiveness of BEACHES in practice, we develop a very large-scale integration (VLSI) architecture and provide field-programmable gate array (FPGA) implementation results. Our results show that adaptive channel denoising can be performed at high throughput and in a hardware-friendly manner for massive MU-MIMO mmWave systems with hundreds of antennas.