In reply to Russell Kaye.

Are you using the PayPal Standard that comes with Woo, a plugin, or what exactly? I've been running transactions in the sandbox without SSL without any issue, but that's with API calls.

I think what you may be referring to is the "ssl handshake failure"..?? That's a little bit misleading. It mentions has SSLv3 is no longer supported, and you have to use TLS now. It's all because of the POODLE vulnerability.