I have a popular ... web site ... thousands of pages ... useful information ... good quailty information ... good information ... easy to navigate etc.
I thought we would be able to get included in the DMOZ directory.
Based on your own review and description of the site, I'd be moderately-confident that the site isn't listable.
Because you didn't mention a single thing that meets the ODP criterion for listing.
Now, you could have missed it completely--but I think that's unlikely, since it's your site and you should know what to look for. What's more likely is that you haven't read the ODP policies carefully enough to realize what the volunteers are looking for.
Popularity isn't on the list. Number of pages isn't on the list. Information really isn't on the list. Quality of information isn't on the list, as such--after all, we list websites including crackpot science and fad diets and presidential campaign speeches.
The crux is "unique". Anyone can have thousands of pages chock-full of useful, good quality information, simply by copying a fraction of a percent of wikipedia, the Internet Archive, Gutenberg.org--or a larger portion of any number of smaller sites. Think of it as a potluck--it's about what you bring, not what you eat.
But, if your navigation isn't clear about what you brought, those thousands of pages could work against your credibility. The ODP reviewer won't check every page--will simply sample. But if the first several samples all turn out to be other people's prize dishes, then -- it's time for the reviewer to move along to another site more likely to contain "unique" content up front.
This is not the place to discuss what YOU brought--the place for THAT is on the website itself. Make sure the navigation is organized so that any visitor (including the ODP reviewer) can find which of those thousands of pages were your cooking. If you do that, then it really won't matter how much you browses other people's dishes--you'll get the credit you deserve (nothing)--but nobody will hold it against you either.