The merger of a neutron star (NS) binary may result in the formation of a rapidly spinning magnetar. The magnetar can potentially survive for seconds or longer as a supramassive NS before collapsing to a black hole if, indeed, it collapses at all. During this process, a fraction of the magnetar's rotational energy of similar to 10(53) erg is transferred via magnetic spin-down to the surrounding ejecta. The resulting interaction between the ejecta and the surrounding circumburst medium powers a year-long or greater synchrotron radio transient. We present a search for radio emission with the Very Large Array following nine short-duration gamma-ray bursts (GRBs) at rest-frame times of approximate to 1.3-7.6 yr after the bursts, focusing on those events that exhibit early-time excess X-ray emission that may signify the presence of magnetars. We place upper limits of less than or similar to 18-32 mu Jy on the 6.0 GHz radio emission, corresponding to spectral luminosities of less than or similar to(0.05-8.3) x 10(39) erg s(-1). Comparing these limits to the predicted radio emission from a long-lived remnant and incorporating measurements of the circumburst densities from broadband modeling of short GRB afterglows, we rule out a stable magnetar with an energy of 10(53) erg for half of the events in our sample. A supramassive remnant that injects a lower rotational energy of 10(52) erg is ruled out for a single event, GRB 050724A. This study represents the deepest and most extensive search for long-term radio emission following short GRBs to date, and thus the most stringent limits placed on the physical properties of magnetars associated with short GRBs from radio observations.