You could use the HTTP::Size module. Take a look at its POD for the get_sizes() method. The POD states that method "fetches all of the images then sums the sizes of the original page and image sizes. It returns a total download size."

Here's an untested example:

use HTTP::Size;
my $total = 
    HTTP::Size::get_sizes( 'http://www.perlmonks.org' );
print "$total\n";
[download]

Hope this helps...

UPDATE: Ok, now I've installed the module and tested the snippet above. ...it works, and at least on one of the test-runs the total size of the Perlmonks front page was 87652 bytes. That will change depending on the amount of chatterbox text, the number of and size of Front Paged articles, etc.