Temporary failure resolving DigitalOcean mirrors

I’ve been working with this droplet (Ubuntu 16.04) for a year and a half now, and all of a sudden after installing and trying to configure tightvncserver everything has gone berserk. the droplet has no access to the internet whatsoever and I’ve tried literally everything people have posted about the same errors without any difference. For instance when I run apt-get update I get the errors:

Temporary failure resolving 'mirrors.digitalocean.com'
Temporary failure resolving 'security.ubuntu.com'
...
W: Failed to fetch http://mirrors.digitalocean.com/blablabla
W: Failed to fetch http://mirrors.digitalocean.com/blablabla
W: Failed to fetch http://mirrors.digitalocean.com/blablabla
W: Failed to fetch http://mirrors.digitalocean.com/blablabla

I can’t even access the log files directly to copy paste the errors, but they are exactly the same as this post’s.

I’m also running an Apache2 server, Go server, MongoDB and MySQL databases on this droplet. And again, everything was working smooth as butter until I installed and tried configuring tightvncserver so I could start working on my project directly on the droplet instead of through FTP… was this not a good idea?..