I was obsessed with this kind of stop motion when I was a young teenager in the 1970s. I used my dad's Super 8 film camera with a cable release so I could fire it one frame at a time. These days, you can do this with basically any DSLR or even a compact camera so long as you can set it to manual exposure and put it on a tripod.
What I am pretty sure was the technique used for most of the film above was to set up the camera on a tripod, frame the subject then lock the tripod down. Shoot a frame, then move the whole tripod back a few inches and have the actor move the same distance. Shoot another frame, move, shoot, move, etc. For some of the later shots where the camera was stationary, the same technique can still be used -- move the subject, shoot, move, shoot, etc. It takes a lot of patience!
The reason for manual exposure is to prevent flickering from frame to frame as the camera adjusts itself to the scene.
Finally, you can usually just import all the frames into your editing software as an image sequence and then set a frame rate. You might find that setting a slower than 24fps rate will work best -- my guess is the video did that too. Back in my Super 8 days I would often click twice for each movement, so 12fps would probably work well.
Give it a try -- one thing is certain: you will have a lot of fun doing it! :)