Note: your system probably requires you to use 'sudo' in front of your privilege access commands, or need to issue 'su root' to gain full root access. Try

sudo cat /etc/resolve.conf 

So, reboot cleared out your DNS cache and now your DNS servers have gone dark. Can you ping IP address 8.8.8.8 or 8.8.4.4 ?

I always log in with root, but tried sudo cat /etc/resolve.conf anyway. The result is the same "No such file or directory".

I can ping both 8.8.8.8 and 8.8.4.4. I can also ping the gateway.

One additional note though, I was playing around with ufw before restarting the droplet. However, after discovering this issue, I have inactivated it, but the problem persists.

@medowlock - Does your system have a /var/run/resolvconf/resolv.conf
If it's there, for diagnostic purpose, create a link to it in etc

ln -s /var/run/resolvconf/resolv.conf /etc/resolv.conf

Then try ping digitalocean.com

This is where it gets really weird, at least for me. When listing the contents of /etc I can see there is a file resolv.conf symlinked to /run/resolvconf/resolv.conf. However

root@ubuntu-512mb-fra1-01:/run# cd /run/resolvconf
-bash: cd: /run/resolvconf: No such file or directory

And running the command you mentioned gives

root@ubuntu-512mb-fra1-01:~# ln -s /var/run/resolvconf/resolv.conf /etc/resolv.conf
ln: failed to create symbolic link ‘/etc/resolv.conf’: File exists
root@ubuntu-512mb-fra1-01:~# cat /etc/resolv.conf
cat: /etc/resolv.conf: No such file or directory

root@ubuntu-512mb-fra1-01:~# ping digitalocean.com
ping: unknown host digitalocean.com

Thanks for all your answers and patience!

@medowlock - Oops, need to remove the existing file before the link command:

mv /etc/resolv.conf /etc/resolv.conf.hold
ln -s /var/run/resolvconf/resolv.conf /etc/resolv.conf

@gndo No luck.

But I did manage to get it working again. What I did was remove the symlink altogether and created the resolv.conf file manually and added to its contents:

nameserver 8.8.8.8
nameserver 8.8.4.4

@medowlock - So that means that the program (resolvconf) that dynamically creates that file, had failed and probably wrote something in your system logs. At some point you will want to look for that and fix the root cause since a future update could mess you up again.