We study the problem of global stabilization by smooth output feedback, for a class of n-dimensional homogeneous systems whose Jacobian linearization is neither controllable nor observable. A new output feedback control scheme is proposed for the explicit design of both homogeneous observers and controllers. While the smooth state feedback control law is constructed based on the tool of adding a power integrator, the observer design is new and carried out by developing a machinery, which assigns the observer gains one-by-one, in an iterative manner. Such design philosophy is fundamentally different from that of the traditional "Luenberger" or "high-gain" observer in which the observer gain is determined by observability in one step. For a class of high-order non-homogeneous systems, we further show how the new design method, with a suitable twist, can still achieve global output feedback stabilization.