I saw a gif of this earlier this week!
I wanted to learn how to make a biped robot from scratch a few years ago. I even have the plans somewhere. You'd need a good few servos (one for each joint), a microcontroller, good set of batteries, probably a bluetooth or wifi adapter, and you'd want a bunch of sensors so it can interact with the outside world. I could see it getting expensive, especially if you want to make it look good.
I think the really hard part would be programming it. Getting the servos to work together to create walking would probably be difficult. Even those robots do a crab walk instead of a normal one. Balance is hard.
I still want to do this someday lol. I forget a lot of the hardware I've learned in school though... I've become a soft, squishy Java developer.