In this paper, we propose a data-driven approach to robust feedback controller design for unknown linear time-invariant (LTI) dynamic systems. Using input-state trajectories and prior knowledge of unknown-but-bounded disturbances, the objective is to synthesize a state-feedback controller that achieves robust stabilization and $\mathcal{H}2$ performance while employing a common quadratic Lyapunov function. Previous works have exclusively considered bounded disturbances described by quadratic matrix inequalities (QMIs) and pointwise $\ell_2$ or $\ell\infty$ constraints. In contrast, this paper introduces a more general framework that characterizes disturbance bounds using compact basic semi-algebraic (BSA) sets, thereby capturing both time-domain and frequency-domain properties. We cast the necessary and sufficient conditions for quadratic stabilization and $\mathcal{H}_2$ performance as convex sum-of-squares (SOS) optimization problems. Additionally, we propose relaxation methods to reduce computational complexity by leveraging the geometric and structural properties of the polynomials defining the BSA sets. Simulation results demonstrate the efficiency and flexibility of the proposed approach.