So, basically, nothing that's supernatural/magical/fantastical/etc... Something that's based on the real world as we know it.
Some of the best stories in games that I've encountered are the BioShocks, Grim Fandango, and Planescape: Torment. But those games - and most other great game stories - involve all manner on surreal things (not that it's a bad thing). However, I would like to find some good story-based games that try to tell an interesting story within the confines of reality (realistic people, realistic settings, realistic scenarios).
I am going to preempt people that are going to suggest Spec Ops: The Line by mentioning that I've already played it :P