We report on our Chandra and Rossi X-Ray Timing Explorer (RXTE) observations of the bright old nova V603 Aq1 performed in 2001 April and supplement it with our analysis of archival X-ray data on this object. We find that the RXTE data are contaminated by the Galactic ridge X-ray emission. After accounting for this effect, we find a high level of aperiodic variability in the RXTE data, at a level consistent with the uncontaminated Chandra data. The Chandra High Energy Transmission Grating spectrum clearly originates in a multitemperature plasma. We constrain the possible emission measure distribution of the plasma through a combination of global and local fits. The X-ray luminosity and the spectral shape of V603 Aq1 resemble those of SS Cyg when it is in transition between quiescence and outburst. The fact that the X-ray flux variability is only weakly energy dependent can be interpreted by supposing that the variability is due to changes in the maximum temperature of the plasma. The plasma density is likely to be high, and the emission region is likely to be compact. Finally, the apparent overabundance of Ne is consistent with V603 Aq1 being a young system.