Using first-principles density-functional calculations, we explore the possibility of magnetic order at the rebonded DB step of the Si(001) surface. The rebonded DB 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.