package Module::Monkey;
use base 'Module';
BEGIN {
  Module->import();
}
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Note that use base does NOT call import() on the superclass.

You're trying to use Module as a base class, and as a holder of constants to be exported, and as a "module loader" that loads all its subclasses. Life would be a lot simpler if you would drop at least one of these functions in another module, and I would recommend splitting them all out, which will make your code a lot saner:

# in Module/Loader.pm
package Module::Loader;
use Module;
use Module::Monkey;
use Module::Ape;
use Module::Constants;

# in Module.pm
package Module;
use Module::Constants;
sub some_inhereted_sub { };

# in Module/Constants.pm;
package Module::Constants;
use constant {
    MDB_INT        => 0,
    MDB_FLOAT    => 1,
    MDB_STRING    => 2,
};

# in Module/Ape;
package Module::Ape;
use Module::Constants;
use base Module;
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Etc.

Hope this helps,
Joost.