You're trying to store the MLDBM::Sync object inside of MLDBM::Sync? Don't do that! Here's the sort of thing I had in mind:

package Calendar;

# constructor
sub new {
    my $class = shift;
    my %args = @_;
    
    my $calendar = {};
    my %data;

    my $sync_dbm_obj = tie(%data, 'MLDBM::Sync', "$args{DIR}\\$arg
+s{year}.db", O_CREAT|O_RDWR, 0640) 
                        or die "can't open tie to $args{year}.db: $!";
    
    $calendar->{'data'} = \%data;

    # if just created, initialize hash with default values
    if( !$calendar->{data}{months} ) {

        $sync_dbm_obj->Lock;

        $calendar->{data}{year} = $args{year};
        $calendar->{data}{months} = {};
        my $aref = $calendar->{data}{months};
        foreach my $month (1..12) {
            $aref->{$month} = _init_days($calendar->{data}{year}, $mon
+th);
        }
        $calendar->{data}{months} = $aref;

        $sync_dbm_obj->UnLock;
    }
    
    return bless $calendar, $class;
}
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When you need access to the MLDBM::Sync object you just say my $sync_dbm_obj = tied $calendar->{data} to get it.

Of course this is not an OO approach. If you really want to use OO, you should make accessor methods for $calendar->months(), $calendar->year(), etc. instead of just using a tied hash.