This paper considers real-time control and learning problems for finite-dimensional linear systems under binary-valued and randomly disturbed output observations, which arise in various fields such as the information industry and control engineering. This has long been regarded as an open problem because the exact values of the traditional regression vectors used in the construction of adaptive algorithms are unavailable, as one only has binary-valued output information. To overcome this difficulty, we consider the adaptive estimation problem of the corresponding infinite-impulse-response (IIR) dynamical systems and apply the double array martingale theory that has not been previously used in adaptive control. This enables us to establish global convergence results for both the adaptive prediction regret and the parameter estimation error, without resorting to such stringent data conditions as persistent excitation and bounded system signals that have been used in almost all existing related literature. Based on this, an adaptive control law will be designed that can effectively combine adaptive learning and feedback control. Finally, we are able to show that for any given bounded reference signal, the closed-loop adaptive control system is globally stable and the long-run average output tracking error tends to zero as time goes to infinity. To the best of the authors' knowledge, this appears to be the first adaptive control result for general linear systems with general binary sensors and arbitrarily given bounded reference signals.