Lighting for shots such as this one is not about absolute lighting levels, it is about lighting ratios. The 'dark' parts of the picture don't need to be in absolute darkness. They just need to be less bright than the bright parts of the scene by enough to make them dark by controlling exposure.
As long as the part of the scene you want illuminated is lit brightly enough compared to the part you don't want to be bright and exposure is set appropriately you'll get the effect you are looking for.
This image was taken in fairly pedestrian indoor lighting levels. There was a black curtain about 15 feet behind the subject. A fairly hot flash was illuminating the subject at about a 45° angle to lower camera left.
ISO 100, f/4, 1/80 second. No overall brightness adjustments in post. Contrast was increased +2, but the shadows were also brightened +2 and the highlights pulled back -3
Here is the wider scene at the same exposure setting for a test shot before the curtain was adjusted to cover the background behind the subject. The ambient light is underexposed by about 4 stops.
Here's an uncropped version of the first image above with the exposure pushed three stops in the raw conversion. At that point the house lights had been cut, but the overhead stage lights were the same as in the test shot. You can see the flash at lower left is spilling onto the curtain a bit, but the difference between the brightness of the curtain and the face is enough to leave the curtain very dark at the selected exposure settings as seen in the crop above.
When you want the background to be dark the key is to get enough distance between your subject and the background so that light illuminating the parts of the scene you want to be bright does not spill onto the background you want to be darker. The same is true of walls off to camera left and right: they need to be far enough away and non-reflective enough that light won't bounce off of them and back into the scene.
With a dark background you can flag your off camera flash to reduce light spilling onto the background if you need to.
For how this works out with a dark background, please see: If I want to shoot darker shots, is it better to increase the lighting and then darken in post to retain clarity? Note that the dark areas in the photo can be much brighter than they look in the image, exposure just has to be set to 'kill the ambient' and the parts you want to be bright have to be lit brightly enough to still be bright at that reduced exposure value.
For another question that deals with a lighter background, please see: How to get this particular black and white look?