It can be done either in-camera or in post, but the strongest results usually start during the shoot.

Common ways to get a gray/black background:

• use a dark backdrop
• keep the background far behind the subject so it receives less light (inverse square law)
• control spill with flags, grids, barn doors, or black cards
• lower ambient light so the background doesn’t brighten up
• expose for the subject while letting the background fall darker

A gray background is easier: even a white or colored background can render gray in a black-and-white image if it’s lit less than the subject.

A deep black background requires tighter lighting control. You often need to “kill the ambient” and prevent your lights from reaching the background.

Post-processing can also do it:

• convert to monochrome
• darken or replace the background with masking
• shift tones/colors to darker gray or black

For your examples, the monochrome processing is clearly part of the look, but the lighting setup during capture is still important. In short: shoot for a dark background first, then refine it in post.