Running youtube cost's around 700 million dollars per year. And I don't think they're profiting very much from youtube.
I believe I read somewhere awhile back that youtube was only making 240 million in revenue from the advertisement's.
Not sure how much youtube red subscriptions are bringing in.
Youtube operates it's own data centers all around the world and has several copies of the same videos on their web servers to serve the video to you from their closest data center to help with the speed of delivery of the video.