You can do it easily by setting the attribute as readonly but providing a private writer:

#!/usr/bin/perl
use Modern::Perl;

{
    package Foo;
    use Moose;

    has bar => (
        is     => 'ro',
        writer => '_set_bar',
    );

}

my $foo = Foo->new;

$foo->bar('something');

# outputs:
# Cannot assign a value to a read-only accessor at
# /home/brunov/tmp/pm.pl line 16

$foo->_set_bar('something');
say $foo->bar;

# outputs:
# something
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Nothing stops your class' consumers from setting bar using _set_bar; they should follow the convention that methods with preceding underscores are private.