We have observed at high resolution (lambda/DELTA-lambda = 5 x 10(4)) and good signal-to-noise ratio (between 50 and 250, with the majority having S/N greater than or similar to 100) a group of 18 early-type stars in the solar neighborhood in the region of the K I lambda-7699 resonance line, to detect interstellar absorptions due to K I. Using our detections, and upper limits, we measure K I column densities and estimate those of H I along these lines of sight. Selecting estimates of interstellar absorption in K I and also in other species, from literature observations in the same angular range centered on l = 152-degrees, b = -9-degrees, we examine a conical volume of semiangle approximately 20-degrees out to some 200 pc from the Sun, placing constraints on a number of the individual cold clouds (T less than or similar to 100 K) in this zone. The technique entails estimating the H I number density of a cloud using the H I column density, and combining it with the cloud radius estimated by identifying neighboring lines of sight in which the cloud is or is not present. The interstellar absorptions allow us to "tag" a given cloud via its radial velocity, to within +/- 1.5 km s-1. Using the hypothesis of pressure confinement, we find temperatures in these clouds of less than or similar to 100 K; as a typical cloud mass ranges up to a few hundred solar masses, which is lower than the estimated Jeans mass, pressure confinement is justified. We also show that the mass fraction of the local interstellar medium (LISM) in these cold clouds is high, at least 80% of the total, while the filling factor is low, well below 10% of the total LISM volume. We detect one cloud with higher densities and lower temperatures, which the physical conditions suggest is on the edge of atomic-molecular equilibrium.