This specificity can relate to subject matter. For example, if you have a site about comedy, you could focus on performance as opposed to writing. You can also target a particular geographic area. I have done this a few times and it seems to help quite a bit. (This site is an example of this approach.)
If Google gets a sense that you're aiming at a particular geographic area, and it has enough data to see that people from there seem to like your stuff (eg, low bounce rate, etc) then that will help a lot.
Here's an example: I've got a website called Blog Plus Tweet. It's kind of specific in its focus. It's about blogging and social media mainly. Now, I've got a lot of blog content up on it now. It's been going a while. But it gets very little search engine traffic.
Now this has a lot to do with the fact that it's a saturated niche anyway. It falls into that whole "blogging about blogging" niche, after all. But there's also the fact that it isn't aimed at any particular geographic area.
I have another site that I have built to promote a couple of services I offer to people in my city of Perth. It's called Perth Business Help.
Now, there are several factors that make it very clear who this site is for. Firstly, there's a city keyword in the domain name. It also has the .au extension. I've also put a fair amount of Perth specific content in there such as names of locations, suburbs, local businesses etc. I've linked out to some local websites. So Google has really got a sense of the "Perthiness" of this website.
What I've discovered is that I'm getting heaps more search engine traffic than the older Blog Plus Tweet. And its not just via searches that contain local keywords. It's doing alright for some generic searches as well. It's displaying the relevant pages only to local searchers, though.
So, before you build a site, think long and hard about the location of the people you want to reach. If it has to be global, fine. Just narrow its focus somewhat so that you've got a better chance of ranking for the content. And if you can make it city or country specific in some way without betraying the whole site's purpose, even better. Odds are you'll have more success search engine wise.