This should get you started, we don't know before hand whether a given line has the log entry that we need to capture but we assume that the log entries follow a certain order that may be intermittent by other uninteresting log entries (i.e each new log record starts with the 'System' entry). In the code below I am capturing the server name, the bytes in and the bytes out into a hash of incremental records, you can extend on that to capture the rest of the entries you seek..

#!/usr/local/bin/perl

use strict;
use warnings;
use Data::Dump qw(pp);

my %log;
my $record = 0; #to be incremented on each round
while(my $line = <DATA>){
        chomp $line;
        my ($system,$server,$b_in,$byteIn,$b_out, $byteOut);
        if($line =~ /System/){
                $record++ ;
                ($system,$server) = split / : /,$line;
                push @{$log{'record'.$record}},{$system=>$server}
                }elsif($line =~ /ByteIn/){
                        ($b_in, $byteIn) = split / : /,$line;
                        push @{$log{'record'.$record}},{$b_in=>$byteIn
+} ;
                        }elsif($line =~ /Byteout/){
                                ($b_out, $byteOut) = split / : /,$line
+;
                                push @{$log{'record'.$record}},{$b_out
+=>$byteOut}
                                }

        }

print pp \%log;
__DATA__
System : system001-server
Box : HW a2b 1234Mb 123.4, address is 0012.d345.1234 (abc 0012.d123 +.
+4567)
Byteout : 5816943852464
ByteIn : 4
Description : example_system_server1_example
System : system002-server
Box : HW a2b 1234Mb 123.4,
ByteIn : 3385
Byteout : 58169
Description : example_system_server2_example
[download]

OUTPUT

{
  record1 => [
               { System => "system001-server" },
               { Byteout => "5816943852464" },
               { ByteIn => 4 },
             ],
  record2 => [
               { System => "system002-server" },
               { ByteIn => 3385 },
               { Byteout => 58169 },
             ],
}
[download]

read Data::Dump, Data Structures Cookbook (perldsc), perlref