Camera quality?

To a certain degree, yes. A camera with better UI allows you to explicitly control things like the ISO setting, focus point, aperture, and shutter speed for what you want.

Lighting?

In the first shot, yes. In the second shot, it's more about making the most of the light that's there. Adding more light to a scene can allow you to use a lens stopped down, which gives you more DoF and is probably closer to where the lens performs well. In the second shot, it may also be a really good lens, which is not something you've considered in your list.

But mostly, if you light well, you can control the placement and quality of light to get a specific mood and look in an image.

Post-processing?

Again, yes. Knowing how to properly sharpen without creating artifacts can also help a lot.

Luck?

No, not really. Or the kind of luck that a photographer learns to make for themselves by being ready when the great image comes along or to stick with it until you make that image happen no matter what (have you stopped and made a stranger pose for you, today?). Portrait photography is about people. So you have to know how to interact with and connect to a subject. Not to mention getting their permission for you to point your camera their way.

You have to know your camera, know your settings, and be able to make the opportunities to take the shot. If you can't do that, then you'll always be "unlucky". They say it takes 10,000 hours of practice to get competent at any given skill. The photographers who took those images have probably got more than twice that time under their belts.

How would I go about if I want to take a photo like one of these?

Get a decent camera with RAW, full Manual (M) mode, and a flash hotshoe. If it has an interchangeable lens mount, get a portrait lens for headshots; a 50e or 85e f/1.8 prime, maybe. Get a RAW conversion application for your computer; learn to shoot RAW and post-process. And learn to light (also to deconstruct lighting of images you see). And put in those 10,000 hours. And work on those people skillz. That's probably all in reverse order of importance.