In this time and age, memory is very cheap. What used to cost $300 for a 16MB (yes, I did mean mega-byte) hard-drive, you can get a 2GB in device the size of a keychain (or smaller) for $40.
YouTube is owned by the powerhouse company, Google. They probably have a room (or building) that is dedicated to storage... many servers with multiple terrabyte drives (that's 1024 GB) to hold all the new videos that people are posting every day.
So, I don't think they will be running out of space any time soon.