Youtube data and the Youtube videos that people upload are stored in huge data centers on thousands of web servers.
Google own Youtube and so the Youtube videos and data are stored in Google Data centers which are spread across different states in the USA.
Also Google has some data centers in other countries as well so that you get fast download and streaming speeds from Youtube.
Googles own data centers for Google.com such as searches, indexing etc are all stored in Googles data centers as well.
Google data centers are in eight locations in the U.S., one in South America, four in Europe and two in Asia.
Googles cloud sites, however, are expanding, and Google's cloud map shows many points of presence worldwide.
Google has billions of petabites of data stored so they need lots of storage.
When you upload a video to youtube the video is copied to more than one server so that you don't lose the video should one server or hard drive crash.