[derobert]

For it being essentially impossible, its sure been done a lot: Netflix, MindGeek (NSFW), and Amazon definitely have at least as large video streaming (especially with Amazon now owning Twitch). There are a lot of large, though not really global, ones too (e.g., in the US there is Hulu, CBS, HBO, various sports, etc.). I'm not sure how big Facebook's video streaming (probably fairly large, considering its Facebook). Smaller YouTube-like sites exist; for example Vimeo is still around.

Building a video hosting site costs a lot less than billions. Unless you have YouTube's traffic, you don't need their capacity.