A new method is used to measure the physical conditions of the gas in damped Ly{alpha} systems (DLAs). Using high-resolution absorption spectra of a sample of 80 DLAs, we are able to measure the ratio of the upper and lower fine-structure levels of the ground state of C^+^ and Si^+^. These ratios are determined solely by the physical conditions of the gas. We explore the allowed physical parameter space using a Monte Carlo Markov chain method to constrain simultaneously the temperature, neutral hydrogen density, and electron density of each DLA. The results indicate that at least 5% of all DLAs have the bulk of their gas in a dense, cold phase with typical densities of ~100/cm3 and temperatures below 500K. We further find that the typical pressure of DLAs in our sample is log(P/k_B_)=3.4(K/cm3), which is comparable to the pressure of the local interstellar medium (ISM), and that the components containing the bulk of the neutral gas can be quite small with absorption sizes as small as a few parsecs. We show that the majority of the systems are consistent with having densities significantly higher than expected for a purely canonical warm neutral medium, indicating that significant quantities of dense gas (i.e., n_H_>0.1/cm3) are required to match observations. Finally, we identify eight systems with positive detections of Si II*. These systems have pressures (P/k_B_) in excess of 20000K/cm3, which suggest that these systems tag a highly turbulent ISM in young, star-forming galaxies.