Is there an internet service available that can show me the number of
hits on a non-existent webpage?

In general it's difficult if not impossible to tell the number of hits
on a page for which you don't have access to the server log (some
methods like network traffic interception exist but aren't commonly
used).

Assuming you have access to the domain server log, you can examine
that to get the required information.

It is possible to find that out through your application.  It is all
depends on what language/script that you used to build your
application.

For example, using Java/J2EE, we have the control to see which request
it is and do count on the 404 error page.

So probably you could give us what exactly you are using, i guess,
that could help us in telling you what it is..

If you're on an apache server and can create .htaccess files, you
could try redirecting all requests to some logging script using
mod_rewrite.  See the url below for more info:

http://httpd.apache.org/docs/1.3/mod/mod_rewrite.html

Various scripts like wordpress use mod_rewrite for other purposes like
shorter/more friendly urls (i.e. changing
www.sitename.com/index.php?contentid=thispage" to
"www.sitename.com/thispage.html") - but the basic concept of
redirecting all requests to a single script is the same.

It would take some doing, however you could probably write a pretty
elaborate script to parse any url down to the base files that are
being requested and log whether or not they exist before passing the
user on to either a custom error page, or a generic 404.

Speaking of custom error messages - you might be able to just create
error pages that create an entry in some log any time a 404 occurs.