This is allegedly targeted at the root servers. It doesn't matter what DNS provider you use if your TLD is offline.

Delegations from the root to the TLDs and from the TLDs to second-level domains generally have TTLs of over 24 hours. This attack will have zero effect.

Good point. They'll bring it down for a day and nothing will happen.

Is it likely that the big email providers keep the caching longer and could maintain some functionality in the event the TLD nameservers are down for more than 24 hours?