As most all have said you can't just pop out of the terminal and go into the city. That, by the way, is true of all international lay overs. You have to go through security, customs, and passport/visa whenever you leave the international terminal...even if you don't leave the airport.
But if your baggage is checked straight through to your destination, you don't even have your luggage let alone have to check it through. But you will have to go through all the lines to get back into the international terminal again.
So I don't think going into Beijing during a layover can be done (the visa) but even if it could, it's probably not worth the hassle. So I recommend you just enjoy their new terminal. When my wife and I went to Beijing the old terminal was still in operation; so we didn't have that luxury.
However if you have a chance one day, Beijing is a great city to tour. Our hotel was in the central part of the city; so we opted to just walk around and see the downtown on our own. And, yes, contrary to some answers, you can do that. They don't watch your every step...that's just a myth.
As to where to eat...the Hard Rock Cafe. Yeah, they have one in Beijing. My wife and I went there one evening because, get this, we were tired of Chinese food. In fact during our five night stay in Beijing we saw KFC, Hard Rock, Baskin Robbins, and McDonald's all around the city.