The best multistage pumps can reach ~0.0001 mbar.[0]
Generally, attainable vacuum by pumping depends on leakage rate vs pump speed, and also on the vapor pressures of components. Such as oil used for lubrication, grease for sealing, rubber or plastic hoses, etc.
So for a high vacuum system, you want just metal and glass, and you want them to be very clean. Fingerprints are a huge issue. Also, you typically put a cold trap between your pump and the working volume. Both to get lower pressure, and to avoid contamination. For low vacuum, dry ice in acetone is OK. Then liquid nitrogen, and finally liquid helium.
0) https://cds.cern.ch/record/1046850/files/p43.pdf
Edit: It's important to avoid contamination of the working volume (say, by pump oil) and of the pump (say, by water, or corrosive substances).