The polymeric properties of DNA molecules, which are electrostatically bound to glass-supported cationic lipid membranes, are investigated. The electrostatic interaction is sufficiently strong to hold DNA flat onto the fluid lipid surface but allows DNA to diffuse freely in-plane. The molecules are fluorescently labeled, and fluorescence images are examined in terms of real-space monomer distributions of polymer chains. The chain extension of single DNA fragments of restriction enzyme digests shows power law scaling with number of base pairs in accordance with self-avoiding walks in two dimensions. Dynamic scaling is found for center-of-mass diffusion following Rouse dynamics, D similar to 1/N, and for rotational relaxation times, tau(r) proportional to N-mu with mu = 2.6 +/- 0.4. A crowded surface of monodisperse lambda-DNA behaves like a two-dimensional semidilute solution with a measurable correlation length xi being smaller than for dilute preparations. Polymer unbinding and the role of surface defects are discussed.