{ package MyDate;
use strict;
use warnings;
    sub AUTOLOAD {
        our $AUTOLOAD;
        say "Going into AUTOLOAD";  # Why doesn't this print?
        my $method = $AUTOLOAD;
        $method =~ s/.*:://;
        grep {$method eq $_} qw/yak day month/ or Carp::croak "Invalid
+ method: $method.";
        if ($method eq "day") {
            eval {
                sub day {(localtime)[3];}
            };
        }
        elsif ($method eq "month") {
            eval {
                sub month {(localtime)[4] + 1;}
            };
        }
        Carp::croak $@ if $@;
        goto &$method;
    }
}
######################################################################
+##################
package main;
use strict;
use warnings;
say MyDate->day;        
say MyDate->month;
[download]

String found where operator expected at 879141.pl line 6, near "say "G
+oing into AUTOLOAD""
        (Do you need to predeclare say?)
String found where operator expected at 879141.pl line 9, near "Carp::
+croak "Invalid method: $method.""
        (Do you need to predeclare Carp::croak?)
syntax error at 879141.pl line 6, near "say "Going into AUTOLOAD""
syntax error at 879141.pl line 9, near "Carp::croak "Invalid method: $
+method.""
BEGIN not safe after errors--compilation aborted at 879141.pl line 26.
[download]

The first error tells me that say needs to be imported, the second that Carp isn't loaded. If nothing was printed for you, then that package didn't have warnings enabled.

2: Your code looked like this: if (foo) { eval { sub bar {} } }. You were defining a sub inside the AUTOLOAD sub, and then executing that code by name, instead of assigning to another variable a coderef, like $code = sub { ... }; and then later calling it or using goto on it.

3: Notice that the "was called" statement is executed unconditionally, after it's determined that DESTROY wasn't called.

Actually I get the impression that AUTOLOAD isn't even being called.