In this work, we optimized a clean, versatile, compact source of soft X-ray radiation $(E_{\text{x}\text{-}\text{ray}}\sim 3~\text{keV})$ with an yield per shot up to $7\times 10^{11}~\text{photons}/\text{shot}$ in a plasma generated by the interaction of high-contrast femtosecond laser pulses of relativistic intensity $(I_{\text{las}}\sim 10^{18}{-}10^{19}~\text{W}/\text{cm}^{2})$ with supersonic argon gas jets. Using high-resolution X-ray spectroscopy approaches, the dependence of main characteristics (temperature, density and ionization composition) and the emission efficiency of the X-ray source on laser pulse parameters and properties of the gas medium was studied. The optimal conditions, when the X-ray photon yield reached a maximum value, have been found when the argon plasma has an electron temperature of $T_{\text{e}}\sim 185~\text{eV}$, an electron density of $N_{\text{e}}\sim 7\times 10^{20}~\text{cm}^{-3}$ and an average charge of $Z\sim 14$. In such a plasma, a coefficient of conversion to soft X-ray radiation with energies $E_{\text{x}\text{-}\text{ray}}\sim 3.1\;(\pm 0.2)~\text{keV}$ reaches $8.57\times 10^{-5}$, and no processes leading to the acceleration of electrons to MeV energies occur. It was found that the efficiency of the X-ray emission of this plasma source is mainly determined by the focusing geometry. We confirmed experimentally that the angular distribution of the X-ray radiation is isotropic, and its intensity linearly depends on the energy of the laser pulse, which was varied in the range of 50–280 mJ. We also found that the yield of X-ray photons can be notably increased by, for example, choosing the optimal laser pulse duration and the inlet pressure of the gas jet.