The errors shown indicate that there is a network or DNS issue here. Have you changed the DNS servers listed in /etc/resolv.conf?

I would take the following steps to troubleshoot this:

• Check that you can reach other outside services. Try a ping of security.ubuntu.com (Ctrl+c will exit the ping command)

• If the ping fails, check that you can ping an IP address, try 8.8.8.8 (Google public DNS server)

• If the IP address ping fails, open a ticket with our support team as there may be an issue with your droplet’s network connection (especially if you are doing this from the console).

• If that ping works, check the file /etc/resolv.conf and use that 8.8.8.8 address as a nameserver by adding the line (if it doesn’t exist):

nameserver 8.8.8.8

I hope these steps help you to find a solution. If you get it resolved, let us know what worked.

Correct nameserver in /etc/resolv.conf and ping works for security.ubuntu.com and 8.8.8.8 - However still struggling with

W: The repository 'http://mirrors.digitalocean.com/ubuntu xenial Release' does not have a Release file.
N: Data from such a repository can't be authenticated and is therefore potentially dangerous to use.

on sudo apt-get update

If you run into this issue remember to check, amongst other things, that you have not configured UFW or the DO Cloud Firewall to block outgoing http/https. I had done this accidentally and only realized it when I tried to do apt-get install for something

Same problem as well. How can we solve this?