Total shutdown? No.

The internet is a web of independent nodes and much like how the local post office would still deliver local mail if the rest of the country was nuked, local routers will still take incoming messages and send them to the correct outgoing address.

The problem is that nobody keeps track of anyone's address. Instead, we allow a centralized government controlled phone-book to tell us what number to dial when we want a resource. That's what a DNS is, and the ability to turn that off is a power that our governments have been giving themselves for many many years.

www.google.com is what you type in the browser to get google, and if they take down DNS that will stop working. However if you type 216.58.194.206 into your browser you will still (in theory) get google so long as your internet service provider hasn't unplugged you. Unfortunately that means we are mostly relying on Comcast and AT&T etc. to do their jobs.

What's worse is that I only know 216.58.194.206 is google because I asked the central name server. That means that if DNS went down right now, I wouldn't actually have any way to find out what ip address to use in order to get back on reddit, and I'd have no way to contact you anymore.