Yes—this look can be created either in post or in-camera, depending on how authentic you want the grain to be.

For digital photos, the simplest method is to add grain/noise in an editor. Most apps and programs have a grain or noise tool. In GIMP, for example, you can use Filters → Noise → HSV Noise, then optionally add a little sharpening afterward to bring back edge definition. The balance of added grain plus light sharpening is what helps keep the image gritty without losing all fine detail.

For a true film look, use a high-speed black-and-white film with naturally visible grain, then push it: rate the film at a higher ISO than box speed, underexpose accordingly, and extend development time. That combination increases apparent grain and contrast.

So: digital = easiest and most controllable in post; film = more authentic grain if you want the real analog effect.