Laser plasma produced using a double-stream gas puff target is an intense source of soft x-ray (SXR) and extreme ultraviolet (EUV) radiation, however, without the harmful emission of debris associated with a solid target. Debris-free laser plasma x-ray and EUV sources have been applied in many various applications, including metrology, imaging in a nanoscale, tomography, processing materials, emission and absorption spectroscopy, laboratory astrophysics and astrochemistry, radiobiology, and radiochemistry. In this work, the results of the experimental and theoretical studies on the spatial and temporal evolution of laser plasma produced as a result of irradiation of an argon/helium gas puff target with laser pulses of 1.3 or 6 ns time duration generated with an Nd:YAG laser system are presented. Imaging and spectral measurements of SXR emission from the plasma, created in the double-stream gas puff target, have been performed with the use of an x-ray streak camera. The analysis of the results of spectral measurements, supported by numerical simulations of plasma x-ray emission, allowed the estimation of the plasma electron temperature and its changes over time. Experimental data were compared with the results of theoretical studies performed using a computer model of plasma hydrodynamics. It was shown that plasma expansion is fast enough to reduce the plasma density in the laser focus area during the laser–plasma interaction.