And as for the story, I got an idea:
Why don't we relive the glory days of GlaDOS reign. The main protagonist could be a former Aperture employee, trying to escape while GlaDOS shuts down the whole facility, flooding it with neurotoxin.
In case you like that idea, here's how it could start:
You are asked (by another scientist, not GlaDOS) to participate in the testing of the newly invented portal gun. This could be the first testchamber, which can help the player to get familiar with the controls and cause and effect of portals. Thus the player continues to do tests for the (still human) personal... In the meantime GlaDOS gets activated. She instantly takes over the facility, sealing doors and heating up that deadly neurotoxin. Since it's her first time using it, plus having control of the vast complex is a strain on the supercomputer, it'll take a long time for her to kill everyone. So there won't be a timelimited imposed on the player from the very beginning of the game. But, at that very moment, regular chambers and happy testing will be over once and for all and instead time to escape!
I doubt you can put much story in there, as voice over is highly doubtful, so you should limit reading to a bare minimum... But you could introduce characters, which the player encounters. Employees such as Doug Rattmann might work pretty well. He's the artists of the graffiti in Portal and also a former Aperture employee, one of the few survivors of the GlaDOS' great purge. Read up on him here:
http://sectorw.wikia.com/wiki/Doug_Rattmann
Thats it for now. Good Night!