It's possible to achieve this anywhere without a black backdrop at any time of the day, I've done it on myself in my living room with my TV cabinet in the backdrop, but this method probably needs compositing in order to achieve the same level of effect on everyone.
What you need is an off camera flash and a reflective umbrella on a lightstand. Point the flash in towards the umbrella, and do not fully open the umbrella, in order to narrow down the light coming back out the umbrella.
Meanwhile, set your camera ISO to the lowest setting, and the shutter to the fastest sync speed your flash can support (about 1/180 to 1/200), and start off with a small aperture (maybe about f/11?). Take a test shot without firing the flash. You will want to achieve a completely black picture for this, if it's not black bring down the aperture until it is.
So when you got your ISO, shutter speed and aperture correct you're set. Position your flash, face the person towards the umbrella, and shoot with the flash. The face will be illuminated while the background will be dark. Make adjustments to the flash setting if necessary. Normally I start my flash at about 1/2 to 3/4 power and slowly adjust it until I get the desired amount of lighting.
This worked pretty well for me when I was trying to capture my profile picture for Facebook. It wasn't perfect, but considering I did this at 3am in the morning I wasn't really going to care much about getting the perfect picture anyway. Just make sure that your background doesn't have anything in the vicinity and it shouldn't be lit.