  We present and apply a method to identify the stellar content of the ROSAT
all-sky survey (RASS). We performed a crossmatch between the RASS sources and
stellar candidates selected from Gaia Early Data Release 3 (EDR3) and estimated
stellar probabilities for every RASS source from the geometric properties of
the match and additional properties, namely the X-ray to G-band flux ratio and
the counterpart distances. A comparison with preliminary detections from the
first eROSITA all-sky survey (eRASS1) show that the positional offsets of the
RASS sources are larger than expected from the uncertainties given in the RASS
catalog. From the RASS sources with reliable positional uncertainties, we
identify 28630 (24.9 %) sources as stellar; this is the largest sample of
stellar X-ray sources to date. Directly from the stellar probabilities, we
estimate the completeness and reliability of the sample to be about 93 % and
confirm this value by comparing it to the identification of randomly shifted
RASS sources, preliminary stellar eRASS1 identifications, and results from a
previous identification of RASS sources. Our stellar RASS sources contain
sources of all spectral types and luminosity classes. According to their
position in the color-magnitude diagram, many stellar RASS sources are young
stars with ages of a few $10^7$ yr or binaries. When plotting the X-ray to
bolometric flux ratio as a function of the color, the onset of convection and
the saturation limit are clearly visible. We note that later-type stars reach
continuously higher $F_X/F_{bol}$ values, which is probably due to more
frequent flaring. The color distribution of the stellar RASS sources clearly
differs from the unrelated background sources. We present the three-dimensional
distribution of the stellar RASS sources that shows a clear increase in the
source density near known stellar clusters.
