Only misconfigured servers that still allow SSLv2 connections are vulnerable, apparently. Checking servers at that site is suspicious in that the page isn't actually processing anything(too fast) and is instead spitting out pre-evaluated 'results'. The site gives an 'all clean' to any small domain I try but checking something like ebay.com leads to dozens of entries.
I apologize for sounding skeptical, we're constantly bombarded with fear tactics designed to motivate us to adopt specific beliefs and 'questioning the answer' in this case just doesn't give complete answers. Besides, nothing is secure if it moves data over public connections. Is there a problem? Probably. Does it warrant my changing anything(including attitude)? Nope.
The 'Security researchers' are listed as Nimrod Aviram, Sebastian Schinzel, Juraj Somorovsky, Nadia Heninger, Maik Dankel, Jens Steube, Luke Valenta, David Adrian, J. Alex Halderman, Viktor Dukhovni, Emilia Käsper, Shaanan Cohney, Susanne Engels, Christof Paar, and Yuval Shavitt all of which are associated with the drownattack website.