It would be very hard to do something that actually affects the whole world’s Internet connection, and even then, the effect would be short-lived at best. However, these kinds of brief disruptions could be exactly what the hackers are going for, as their motive remains unclear.
Nobody has made a statement taking credit for the attacks yet, and some see this is an extra factor to worry about. Many hackers take pride in their work and would be happy to claim responsibility for the extra reputation it would give them.
The fact that there is complete silence on that front could imply that a more unusual entity was responsible for the incident, possibly even a state-sponsored organization. But of course, those are wild speculations which don’t really add much to the discussion.
Many people seem a bit nervous about the situation in any case, and there is a strong expectation at the moment that the incident could repeat itself soon. Until then, people can download lists of IP addresses for popular services and websites, so that they will still have a way to connect to those services in case a central DNS provider is taken down again.
These lists have become quite popular on the Internet in the last few days, and there are many places to download a list like that right now, along with instructions on how to use it. Which, for pretty much any modern operating system, is a trivial task for most users.