We construct new families of $mathcal{N}=(0,4)$ $mathrm{AdS}_3times S^2 times tilde S^2 times S^1$ backgrounds fibred over a 2d Riemann surface in Type IIB string theory. These solutions are obtained by extracting the near-horizon limit of D3-brane box configurations, consisting on D3-D5-NS5 branes ending on D5-NS5 background branes. We relate our families of solutions to previous $mathrm{AdS}_3times S^2 times text{CY}_2$ and $mathrm{AdS}_3times S^3 times S^2$ solutions to Type IIA recently constructed in the literature. We construct explicit 2d quiver CFTs associated to the D3-brane box configurations, and check that the central charges match the holographic result. We extend our solutions to include D7-branes, and show that a subclass of these solutions can be interpreted in terms of D3-D5-NS5 defect branes embedded in a 5d fixed point theory. This is explicitly realised by linking our solutions to a 6d domain wall that asymptotes locally the $mathrm{AdS}_6times S^2 times Sigma_2$ solution T-dual to the Brandhuber-Oz vacuum.
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