I experienced an identical issue twice now on two separate installs on two separate environments. The first was on a client's production environment while I was trying to install an extension using Magento Connect. When I plugged in the extension key in step 2, I was presented with this:
error:14077410:SSL routines:SSL23_GET_SERVER_HELLO:sslv3 alert handshake failure
I shook it off and just installed the extension over FTP and continued about my day thinking something might be wrong with Magento Connect.
I then experienced the same issue with a clean install of 188.8.131.52 on a testing server I setup. Unable to proceed with extension install due to the same error.
Regarding SSL: The client's site has a fully verified SSL certificate. My testing server certificate was self-signed.
At this point I'm really just curious as to what causes it as I am able to proceed with FTP/SSH installs of my extension.
Additional Notes: Both instances were using Magento CE 184.108.40.206.