Sounds like the typical impossible "I want to do iterative task X without iterating" request. Use List::Util and map and you can hide most of it, though.

use List::Util qw( sum );
use YAML::Syck qw( Load );

my $hash = Load( do { local $/; <DATA> } );
my $count = sum( map { scalar @{ $hash->{ $_ } } } keys %${hash} );

print $count, "\n";

__END__
foo:
  - A
  - B
  - C
bar:
  - D
  - E
baz:
  - F
  - G
  - H
[download]

Addendum: Just to make things clear: as others have below I'm interpreting "a total count of elements in a Hash of Arrays" to mean "a sum of the count of elements in each arrayref contained in a hash". And just to add more "twaddle", a Ruby version:

require 'yaml'
hash = YAML::load( <<EOT )
foo:
  - A
  - B
  - C
bar:
  - D
  - E
baz:
  - F
  - G
  - H
EOT

count = hash.values.inject(0) { |s,a| s += a.size }
puts count

## O(1) version, akin to ikegami's below
o_1_count = 0
hash.each_value { |a| o_1_count += a.size }

## Duuurh, Hash isa Enumerable so it has an inject itself
or_even_count = hash.inject( 0 ) { | s, (k,v) | s += v.size }
[download]

Addendum Update: Changed Ruby version to use hash.values rather than hash.keys, as seen below, which is actually a cleaner implementation than mine above. Also added (what should be :) an O(1) Ruby implementation. And also added using Hash#inject directly rather than iterating over values.