YouTube runs many geographically distributed datacenters and can each more-or-less independently serve user requests; there's also sufficient redundancy that a full DC can go down and the rest are able to carry the redirected load. When a link goes down like today it's often not user visible as most requests are already served at least from within the same continent. If a whole data center goes down, users might experience higher latency from longer round-trips and colder caches because their traffic is being routed to a DC further away that now has more diverse traffic, but they also won't experience an outage.
This stuff was fun to work on. :)
Those were fun times on the web, when the alternative was curated lists on Yahoo or... whatever the hell Ask Jeeves was aside from a collection of Easter Eggs directed at rude middle schoolers.