  We present an interacting model for the electronic and magnetic behavior of a
strained $(001)$ atomic layer of Sr$_2$FeMoO$_6,$ which shows room-temperature
ferrimagnetism and magnetoresistance with potential spintronics application in
the bulk. We find that the strong spin-orbit coupling in the molybdenum 4$d$
shell gives rise to a robust ferrimagnetic state with an emergent
spin-polarized electronic structure consisting of flat bands and four massive
or massless Dirac dispersions. Based on the spin-wave theory, we demonstrate
that the magnetic order remains intact for a wide range of doping, leading to
the possibility of exploring flat band physics, such as Wigner crystallization
in electron-doped Sr$_{2-x}$La$_{x}$FeMoO$_6.$
