It's a DNS problem. Basically, your computer can't resolve any of Blizzard's domain names to IP addresses.
Blizzard's DNS servers for all their domains are hosted at cerf-d0t-net, which appears to be some kind of corporate AT&T hosting provider.
DNSValidation-d0t-com reports that battle-d0t-net DNS is served by ns-east-d0t-cerf-d0t-net and ns-west-d0t-cerf-d0t-net, both of which are unreachable.
Similarly, a query on cerf-d0t-net itself shows the same thing. Curiously, AT&T's domain is served via DNS hosts on attdns-d0t-com, so that's fine.
Needless to say, I'm sure Blizzard is kicking up a shitstorm about this right now. It's entirely possible that these DNS servers could be getting DDOSed, but you'd think they'd be fairly robust. To be entirely unavailable means something likely broke in a colossal way.
(Sorry - Newb poster. Can't link actual URLs or put actual hostnames in the post
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