Sysadmin here:
It is completely possible to shut down the internet. Disable the root DNS servers and nothing resolves to the correct IP address. There are 13 redundant root dns servers worldwide. This will never happen though for the whole internet but is does allow the ones in control of the root server too heavily censor. It will be selective shutdowns of sites deemed false or misleading by disabling your dns route at the top level.

You can still get to a web server if its domain is blocked by typing the IP address for the server. In most cases though modern web browsers won't allow using ssl without a fully qualified domain so using the ip address means you will have to jump through hoops to make it work (clicking yes I understand the certificate is invalid etc..) and use unencrypted port 80 http instead of 443 https. This means anyone with access to your network to view all of your traffic unencrypted. And this is assuming you have physical access to your server or trust your hosting provider to not block you.
Oh and your ISP can also kill your static IP at any time, so yeah...there is that too.

glps static IP address is 104.20.51.56 but they have good security and don't allow you to get to by going to [link to 104.20.51.56]