Vacancy line defects on the Si(001) surface induced by small traces of Ni are analyzed with a scanning tunneling microscope. The domain structure that is found obeys the size relation derived by Zeppenfeld et al. [Phys. Rev. Lett. 72, 2737 (1994)] for surface systems with long-range interactions, decaying as L-2. From the thermally induced wandering of the vacancy line defects we extract the strength of this long-range repulsive interaction as well as the short-range attractive interaction between the vacancies in adjacent dimer rows.