From your reported output it appears that just implementing the debugging causes a fatal exception to be thrown - am I reading that right? Seems very odd.

Regardless, I do not see anything like the same debugging output that you do:

IO::Socket::SSL version 2.012
DEBUG: .../IO/Socket/SSL.pm:2564: new ctx 11004128
DEBUG: .../IO/Socket/SSL.pm:504: socket not yet connected
DEBUG: .../IO/Socket/SSL.pm:506: socket connected
DEBUG: .../IO/Socket/SSL.pm:528: ssl handshake not started
DEBUG: .../IO/Socket/SSL.pm:561: using SNI with hostname perlmonks.pai
+rsite.com
DEBUG: .../IO/Socket/SSL.pm:596: request OCSP stapling
DEBUG: .../IO/Socket/SSL.pm:629: Net::SSLeay::connect -> 0
DEBUG: .../IO/Socket/SSL.pm:677: connection failed - connect returned 
+0
DEBUG: .../IO/Socket/SSL.pm:1753: SSL connect attempt failed because o
+f handshake problems

DEBUG: .../IO/Socket/SSL.pm:1758: SSL connect attempt failed because o
+f handshake problems error:14094410:SSL routines:SSL3_READ_BYTES:sslv
+3 alert handshake failure
DEBUG: .../IO/Socket/SSL.pm:1742: IO::Socket::IP configuration failed
DEBUG: .../IO/Socket/SSL.pm:2597: free ctx 11004128 open=11004128
DEBUG: .../IO/Socket/SSL.pm:2602: free ctx 11004128 callback
DEBUG: .../IO/Socket/SSL.pm:2609: OK free ctx 11004128
SSL connection with SSLv3 failed
DEBUG: .../IO/Socket/SSL.pm:2564: new ctx 11004128
DEBUG: .../IO/Socket/SSL.pm:504: socket not yet connected
DEBUG: .../IO/Socket/SSL.pm:506: socket connected
DEBUG: .../IO/Socket/SSL.pm:528: ssl handshake not started
DEBUG: .../IO/Socket/SSL.pm:561: using SNI with hostname perlmonks.pai
+rsite.com
DEBUG: .../IO/Socket/SSL.pm:596: request OCSP stapling
DEBUG: .../IO/Socket/SSL.pm:2467: did not get stapled OCSP response
DEBUG: .../IO/Socket/SSL.pm:2420: ok=1 cert=18472720
DEBUG: .../IO/Socket/SSL.pm:2420: ok=1 cert=19090528
DEBUG: .../IO/Socket/SSL.pm:2420: ok=1 cert=19087024
DEBUG: .../IO/Socket/SSL.pm:2420: ok=1 cert=18989936
DEBUG: .../IO/Socket/SSL.pm:1532: scheme=default cert=18989936
DEBUG: .../IO/Socket/SSL.pm:1542: identity=perlmonks.pairsite.com cn=*
+.pairsite.com alt=2 *.pairsite.com 2 pairsite.com
DEBUG: .../IO/Socket/SSL.pm:629: Net::SSLeay::connect -> 1
DEBUG: .../IO/Socket/SSL.pm:684: ssl handshake done
SSL connection with TLSv1_2 set up
DEBUG: .../IO/Socket/SSL.pm:2597: free ctx 11004128 open=11004128
DEBUG: .../IO/Socket/SSL.pm:2602: free ctx 11004128 callback
DEBUG: .../IO/Socket/SSL.pm:2609: OK free ctx 11004128
[download]

From this it very much does appear that my installation is indeed attempting to connect to the site over SSLv3 without client-side error and that it is (as expected) the server which is rejecting this protocol. HTH.