Im looking for practical advice on which hardware to use (not necessarily arduino) that can control this number of motors. Also, Id like to get an advice on what different hardware parts are needed for the project (besides the main board and the motors)
My only experience with electrical boards is from building a RepRap printer (which I know can control 5 motors)
The same code can control a small stepper as a big one assuming that the stepper driver for the big motor also takes step and direction signals which most of them do.
How many steps per second will be required to control all 8 motors? That may become a limiting factor with a 16MHz Arduino. And AFAIK a serial to parallel shift register will "steal" more CPU cycles.
To emulate that cable robot input/output device you're going to need more than just stepper motors. From the video it looks like at a minimum each motor will need a shaft encoder. That means three inputs (Step, Direction,and Enable) and two outputs (quadrature encoder) for each motor for a total of 40 I/O pins. I suggest you start with an Arduino MEGA 2560.