Laser-driven X-rays as probes for high-energy-density physics spans an extremely large parameter space with laser intensities varying by 8 orders of magnitude. We have built and characterized a soft X-ray source driven by a modest intensity laser of 4 × 1013 W/cm2. Emitted X-rays were measured by diamond radiation detectors and a filtered soft X-ray camera. A material-dependence study on Al, Ti, stainless steel alloy 304, Fe, Cu and Sn targets indicated 5-μm-thick Cu foils produced the highest X-ray yield. X-ray emission in the laser direction and emission in the reverse direction depend strongly on the foil material and the thickness due to the opacity and hydrodynamic disassembly time. The time-varying X-ray signals are used to measure the material thinning rate and is found to be ∼1.5 μm/ns for the materials tested implying thermal temperature around 0.6 eV. The X-ray spectra from Cu targets peaks at ∼2 keV with no emission >4 keV and was estimated using images with eight different foil filters. One-dimensional hydrodynamic and spectral calculations using HELIOS-CR provide qualitative agreement with experimental results. Modest intensity lasers can be an excellent source for nanosecond bursts of soft X-rays.