Hi ahackney,
While, I advise that you take vantage of the wisdom and words of kcott, I will say it's not bad to 'eat' on the boilerplate of perl OO intrinsic design.
..I am trying to make the following hash an attribute of the above class...
You could write subroutine, to 'set' and 'get', the value of the hash attribute like this:

{
package My::Sys;
sub new {
   my $class = shift;
   my $self = {
                  coreID => 0,
                  isBpMaskError => 0,
                  isVrrpErrors => 0,
                  isSymPriorityErrors => 0,
                  isSslProfileErrors => 0,
                  isGlobalPortErrors => 0,
                  isVipsinVipGroupsErrors => 0,
                  isVipGroupErrors => 0,
                  isSnatIpErrors => 0,
                  isCodeVersionErrors => 0,
                  aclsMissingFromMaster => {},
   };
   
   bless $self, $class;
   return $self;
}

sub set_acls{
  my $self = shift; 
   my $cot = 0;

   $self->{aclsMissingFromMaster}{"list ".++$cot} = $_ for @_;
}

sub get_acls{
  my $self = shift;
  return $self->{aclsMissingFromMaster};
}

}
use Data::Dumper;
my $sys = My::Sys->new();
my @arr = ([1..4],[5..8],);
# set the attribute
$sys->set_acls(@arr);
{
local $Data::Dumper::Sortkeys = 1;
# get your values here
print Dumper $sys->get_acls();
}
[download]

..you get...

$VAR1 = {
          'list 1' => [
                        1,
                        2,
                        3,
                        4
                      ],
          'list 2' => [
                        5,
                        6,
                        7,
                        8
                      ]
        };
[download]

Just an example for you to see...
You might want to check some other documentation like perlobj, perlootut.