I have been working for quite some time on a website that highly depends on Paypal working properly on it. Without it, it is really useless.
Around 1 in the morning
Try add
CURLOPT_SSL_CIPHER_LIST => 'TLSv1'
to your PPHttpConfig.php
file. I had the same issue with you and spent hours to find the solution. This worked for me.
One of the reasons for this error is older version of PHP and OPENSSL for e.g. PHP 5.3.5 will cause this error.
Solution, update the PHP (recomended >= 7).
This comes up when searching for Magento Error:14077410:SSL Routines:SSL23_GET_SERVER_HELLO:sslv3 Alert Handshake Failure! If you are trying to solve that, here is the link to the guy who originally solved the issue along with a downloadable patch: https://www.dwdonline.com/blog/fix-magento-error14077410ssl-routinesssl23_get_server_hellosslv3-alert-handshake-failure.html It's the same error - just in another software package.
I've had the same problem.
My server is an old version of Ubuntu 12.04
Check sertificates
echo "All certificates in ca-certificates.crt, listed by subject, check for presence of VeriSign's 'Class 3 Public Primary - G5':"
awk -v cmd='openssl x509 -noout -subject' '/BEGIN/{close(cmd)};{print | cmd}' < /etc/ssl/certs/ca-certificates.crt | grep "G5"
I updated openssl.
sudo apt-get install openssl
I could updated curl only by this instruction. http://pavelpolyakov.com/2014/11/17/updating-php-curl-on-ubuntu/ curl 7.47.1 version
And I updated libcurl. http://juniway.blogspot.com/2015/12/curl-48-unknown-option-was-passed-in-to.html