Photography is not about how much gear costs. Its about light.
That is a 4 light setup.
1 hard light on a "Rembrant" position to the right.
2 side lights (one on each side) Looks hard light to me.
1 Background light.
If you do that with 4 incandescent lightbulb, speed lights or studio flashes, dosen't matter.
There is this YongNuo brand that makes cheap flashes but dosen't feel like they are "cheap". They are well constructed. There are some with a slave mode and can be triged by remote control.
We answered a similar question here: Very similar to How can I get a lot of detail in the face, skin as seen in this poster for The Martian?
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We don't know if we cant see the exif data. But we can gess.
what aperture and shutter speed you'd be looking at to get that level of detail
A good camera, the shutter speed is the sync speed I gess (1/125-1/250). If the room is pich black you can take it at 1 second if you are using a flash and it dosen't matter.
A normal aperture on a portrait photo where you need all in focus is f/8-f/11. This depends on the light intensity and Iso settings. If you want an a f/11 aperture... you most likely need flashes.
while leaving plenty of range to adjust levels and sharpening.
Shoot in raw.