We introduce a coarse-grained numerical model that represents a generic DNA hydrogel consisting of Y-shaped building blocks. Each building block comprises three double-stranded DNA arms with single-stranded DNA sticky ends, mimicked by chains of beads and patchy particles, respectively, to allow for an accurate representation of both the basic geometry of the building blocks and the interactions between complementary units. We demonstrate that our coarse-grained model reproduces the correct melting behavior between the complementary ends of the Y-shapes, and their self-assembly into a percolating network. Structural analysis of this network reveals three-dimensional features consistent with a uniform distribution of inter-building-block dihedral angles. When applying an oscillatory shear strain to the percolating system, we show that the system exhibits a linear elastic response when fully connected. We finally discuss to what extent the system's elastic modulus may be controlled by simple changes to the building block complementarity. Our model offers a computationally tractable approach to predicting the structural and mechanical properties of DNA hydrogels made of different types of building blocks.