We investigate the properties of the blackbody spectrum by direct numerical solution of the classical equations of motion of a one-dimensional model that contains the essential general features of the field-matter interaction. Our results, which do not rely on any statistical assumption, show that the classical blackbody spectrum exhibits remarkable properties: (i)a quasistationary state characterized by scaling properties, (ii)consistency with the Stefan-Boltzmann law, and (iii)a high-frequency cutoff. Our Letter is a preliminary step in the understanding of statistical properties of infinite-dimensional systems.