Have you check to see if you are on a Blacklist?
Are you hosted or self managed?
"Connection refused" or "Connection timed out"
These are firewall issues. Most likely, your webhost is blocking access to the MailChimp API servers.
To fix it, you should ask your webhost for the following URL to be whitelisted:
https://*.api.mailchimp.com
403 "Forbidden" or 503 "Service Unavailable"
This means that the IP address that your hosting provider uses for outbound cURL connections made by PHP is blacklisted by the Akamai firewall (because of poor reputation). Akamai is the firewall MailChimp and many other major websites use to protect their servers, which is why this is somewhat harder to fix.
Note that many hosting providers use shared IP addresses, meaning that your website IP is shared with a number of their users. It's possible that another client of your hosting provider did something to get the IP blacklisted, completely out of your control.
When this happens, we recommend you to change your hosting IP address because having a bad reputation at Akamai does not just affect the MailChimp API.