I receive no error. I have since played with things and have come to the conclusion that something is up with Net:HTTPS.

I have attempted a rebuild and got this:

[root@mail t]# pwd
/root/.cpan/build/Net-HTTP-6.03-vnUzN1/t
[root@mail t]# ls
apache-https.t  apache.t  http.t  LIVE_TESTS
[root@mail t]# /usr/bin/perl apache-https.t 
1..8
# Running under perl version 5.008005 for linux
# Current time local: Wed Jul 11 16:26:19 2012
# Current time GMT:   Wed Jul 11 15:26:19 2012
# Using Test.pm version 1.25
# ----------------------------
# 200 OK
# Content-Type: message/http
# Date: Wed, 11 Jul 2012 15:26:20 GMT
# Server: Apache/2.4.1 (Unix) OpenSSL/1.0.0g
# Transfer-Encoding: chunked
#
# TRACE /libwww-perl HTTP/1.1
# Host: www.apache.org
# User-Agent: Mozilla/5.0
# Accept-Language: no,en
# Accept: */*
# 
ok 1
ok 2
ok 3
ok 4
Status read failed:  at /usr/lib/perl5/site_perl/5.8.5/Net/HTTP/Method
+s.pm line 269.
[download]

I am currently messing about with the test scripts to see what's occurring

Tried to login myself, everything worked normally. Net::HTTP 6.00 is installed in my system, and I tried downloading 6.03, all tests succeeded:

$ perl test.pl
Hurrah! Logged in
$ perl apache-https.t
1..8
# Running under perl version 5.014002 for linux
# Current time local: Thu Jul 12 10:31:42 2012
# Current time GMT:   Thu Jul 12 06:31:42 2012
# Using Test.pm version 1.25_02
# ----------------------------
# 200 OK
# Content-Type: message/http
# Date: Thu, 12 Jul 2012 06:31:47 GMT
# Server: Apache/2.4.1 (Unix) OpenSSL/1.0.0g
# Transfer-Encoding: chunked
#
# TRACE /libwww-perl HTTP/1.1
# Host: www.apache.org
# User-Agent: Mozilla/5.0
# Accept-Language: no,en
# Accept: */*
#
ok 1
ok 2
ok 3
ok 4
# ----------------------------
# 200 OK
# Content-Type: message/http
# Date: Thu, 12 Jul 2012 06:31:47 GMT
# Server: Apache/2.4.1 (Unix) OpenSSL/1.0.0g
# Transfer-Encoding: chunked
#
# TRACE /libwww-perl HTTP/1.1
# Host: www.apache.org
# User-Agent: Mozilla/5.0
# Accept-Language: no,en
# Accept: */*
#
ok 5
ok 6
ok 7
ok 8
[download]

Perhaps the problem lies deeper in the system. Probably, it's related to system libssl?
Sorry if my advice was wrong.