This paper proposes a novel approach to analyze and design distributed robust consensus control algorithms for general linear leaderless multiagent systems (MASs) subjected to relative-state constraints or uncertainties represented by a locally or a globally sector-bounded condition. First, we show that the MAS robust consensus design under relative-state constraints or uncertainties is equivalent to the robust stability design under state constraints or uncertainties of a transformed MAS, which has lower dimensions. Next, the transformed MAS under state constraints or uncertainties is reformulated as a networked Lur'e system. By employing the S-procedure and Lyapunov theory, sufficient conditions for robust consensus and the designs of robust consensus controller gain are derived from solutions of distributed linear matrix inequality (LMI) convex problems. Finally, numerical examples are introduced to illustrate the effectiveness of the proposed theoretical approach.