We study scaling properties of the localized eigenstates of the random dimer model in which pairs of local site energies are assigned at random in a one-dimensional disordered tight-binding model. We use both the transfer matrix method and the direct diagonalization of the Hamiltonian in order to find how the localization length of a finite sample scales with the localization length of the infinite system. We derive the scaling law for the localization length and show it to be related to scaling behaviour typical of uncorrelated band random matrix, Anderson and Lloyd models.