In many modules I see people embed the version number in the POD so the reader will know. This can quickly get out of sync with the real version number.

Write a test. Or a little script that either synchronizes, or increments version numbers.

Would it be better/possible to have perldoc extract the version number of a module - when displaying the docs of a module - and show that along the documentation?

Probably not. But you can always do something like (untested):

{no warnings; <<'=pod'}
=pod

our $VERSION = '1.2.3';

=pod

=cut
[download]

The first =pod tricks most, if not all, podlators into thinking POD starts. They're all fooled, and perl knows. The second =pod starts the real POD section. =cut syncs perl and the podlators.