Using first-principles density-functional calculations, we explore the possibility of magnetic order at the rebonded $D_B$ step of the Si(001) surface. The rebonded $D_B$ step containing threefold coordinated Si atoms can be treated as a one-dimensional dangling-bond (DB) wire along the step edge. We find that Si atoms composing the step edge are displaced up and down alternatively due to Jahn-Teller-like distortion, but, if Si dimers on the terrace are passivated by H atoms, the antiferromagnetic (AFM) order can be stabilized at the step edge with a suppression of Jahn-Teller-like distortion. We also find that the energy preference of AFM order over Jahn-Teller-like distortion is enhanced in an oscillatory way as the length of DB wires decreases, showing the so-called quantum size effects. ; Comment: 12 pages, 4 figures, Accepted in Journal of Chemical Physics