Two-dimensional $mathcal{N}=(0,4)$ supersymmetric quiver gauge theories arerealized as D3-brane box configurations (two dimensional intervals) which arebounded by NS5-branes and intersect with D5-branes. The periodic braneconfiguration is mapped to D1-D5-D5$'$ brane system at orbifold singularity viaT-duality. The matter content and interactions are encoded by the$mathcal{N}=(0,4)$ quiver diagrams which are determined by the braneconfigurations. The Abelian gauge anomaly cancellation indicates the presenceof Fermi multiplets at the NS-NS$'$ junction. We also discuss the braneconstruction of $mathcal{N}=(0,4)$ supersymmetric boundary conditions in 3d$mathcal{N}=4$ gauge theories involving two-dimensional boundary degrees offreedom that cancel gauge anomaly.