Sometimes called Bullet Time. The key to his kind of production is the planning and the shooting, not the post. While post is important, if you don't have the right kind of images then you can't accomplish the task.
Do a Google Search for Bullet Time effect and you'll see other ways to kind of fake it without dozens or even hundreds of cameras.
You don't need that many cameras and even if you do, you can rent them/ borrow them from friends & colleagues. The trick is to find the right combination for each move and make techniques like morphing and warping or even 3D reconstruction of the scene part of the plan... That's what they did many times in "The Matrix" movies - map photos from only a few cameras on simple 3D meshes and re-create the moves using rotomation and 3D matchmoving just like in otehr shots they merely warped from one image to another with lots of motion blur...