Update: I found the problem. The error message is completely unhelpful, but if one of the cipher suites supported by SAP isn't available on the server with which you're trying to communicate, then even if you disable SSLv3, it will still fall back to it (for some unknown reason). I managed to get communication working again by enabling TLS_RSA_WITH_AES256_CBC_SHA on the target server. It's a shame, as there's now the possibility that someone could communicate with that server and not be protected by Forward Secrecy, but unfortunately I don't have any choice until SAP get their act together and start supporting things like TLS 1.1 and 1.2.
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