If server does not have support for SSL V3, evident via this error:
:SSL routines:SSL23_GET_SERVER_HELLO:sslv3 alert handshake failure
then the workaround is to override the API URL like so to your wp-config.php file:
Put this code somewhere in the beginning, before the code that includes wp-settings.php, which is usually at the bottom.
That should fix the issue by making the requests NOT use https.