Two-dimensional (2D) half-metallicity and topological states of matter have been the current research focus in condensed matter physics. Herein, we report a novel 2D material named EuOBr monolayer, which can simultaneously show 2D half-metallicity and topological fermions. The material features conducting electronic states in the spin-up channel but a large insulating gap of 4.38 eV in the spin-down channel. In the conducting spin channel, the EuOBr monolayer shows the coexistence of Weyl points and nodal lines near the Fermi energy. The nodal lines in the material contain rich categories including type-I, hybrid, closed, and open nodal lines. The symmetry analysis suggests these nodal lines are protected by the horizontal mirror symmetry, which cannot be broken even spin-orbit coupling (SOC) is included because the ground magnetization direction in the material is out-of-plane [001]. The topological fermions in the EuOBr monolayer are fully spin-polarized, which can be meaningful for future applications in topological spintronic nano-devices.