Analog beamforming greatly reduces the implementation cost of massive antenna transceivers by using only one up/down-conversion chain. However, it incurs a large pilot overhead when used with conventional channel estimation (CE) techniques. This is because these CE techniques involve digital processing, requiring the up/down-conversion chain to be time-multiplexed across the antenna dimensions. This paper introduces a novel CE technique, called continuous analog channel estimation (CACE), that avoids digital processing, enables analog beamforming at the receiver and additionally provides resilience against oscillator phase-noise. By avoiding time-multiplexing of up/down-conversion chains, the CE overhead is reduced significantly and furthermore becomes independent of the number of antenna elements. In CACE, a reference tone is transmitted continuously with the data signals, and the receiver uses the received reference signal as a matched filter for combining the data signals, albeit via analog processing. We propose a receiver architecture for CACE, analyze its performance in the presence of oscillator phase-noise, and derive near-optimal system parameters and power allocation. Transmit beamforming and initial access procedure with CACE are also discussed. Simulations confirm that, in comparison to conventional CE, CACE provides phase-noise resilience and a significant reduction in the CE overhead, while suffering only a small loss in signal-to-interference-plus-noise-ratio.