Paypal can not connect to Sandbox server. Return error 14077410 (sslv3 alert handshake failure)

You'll have to switch the cURL request to use TLS 1.2 in order to use the PayPal sandbox. I'm in the same boat, and there's no way around it, unfortunately. They just activated the change on the sandbox environment a few days ago.

https://devblog.paypal.com/upcoming-security-changes-notice/

I'll add some extra info on this since the first answer doesn't really cover all of the important points.

Paypal has started rolling out some upgrades, the sandbox now requires TLS 1.2 for all requests, and production systems will also require this from June 2016 onwards.

To support this you will need to:

• Ensure your server has OpenSSL 1.0.1 or above (which is when TLS 1.2 support was added).
  openssl version will show you your version number.

• Once you meet that criteria, in your PHP code you can force the SSLVERSION to TLS 1.2 with the following command:

  curl_setopt($curl, CURLOPT_SSLVERSION, 6);

Or if you want a less hacky solution, it is possible to have the correct SSL version kick in automatically during the handshake, where the client and server compare available ciphers to find a common protocol. You appear to be using PHP + curl, so you'll need PHP 5.5.19+ and curl 7.29+ for this to take place.

I have similar problem with error 14077410 and SSL3. I upgrade my PHP server from 5.4 to 7.0 and error disappear.