package MyModule;
use File::Rsync;
my $rsync;
{
    my %rsync_out;
    $rsync=File::Rsync->new( { outfun => &rsync_out,
                               errfun => &rsync_out,
                           } );
    sub rsync_out {
      my ($message, $type)=shift;
      $rsync_out{$type}.="$message\n";
    }

    sub outmsg { $rsync_out{out} }
    sub errmsg { $rsync_out{err} }
}

# ...
# code here has no clue about %rsync_out
# (but it can still see $rsync because I declared 
# it outside the braces.)
# ...

# Access the output using the outmsg and errmsg subs
my $output = outmsg;