At the start of Portal 2, it's clear that time has passed. You've been woken up every few months for physical and mental fitness tests, until something goes wrong and you're left in stasis for much longer. How long is Chell, your character, in stasis? How much time has passed between the end of Portal and the start of Portal 2?
Assuming the announcer was reciting the number of days you had been in stasis, as with your earlier 50-day wake up call, "it has been 9 9 9 9 9..9 9...[static]", it has been at least 273 years, or 27,300 assuming the pause wasn't a broken-record effect.
In "The Final Hours of Portal 2" e-book, Keighley mentions that:
One way to further differentiate Portal and Half-Life was to set the game far in the future—at least 50,000 years.
It doesn't explicitly say that Portal 2 was set in 52,000 AD, but the general point was that the events between the two series are so distant from one another as to not interfere.
No exact number has been stated that I'm aware of. All we know is that "hundreds of years" have passed between the end of Portal 1 and the beginning of Portal 2.
In an unused GLaDOS voice line she states that:
Fifty thousand years is a lot of time to think. About me. About you. We were doing so well together. Source
Referring respectively to herself and the protagonist Chell.
According to the Combine Overwiki, the appearance of wood, paper and upholstery would suggest decades rather than years. Portal took place roughly around the same time as the Combine Invasion of Earth (just after Half-Life [estimated to be 2003]]), Chell was the placed in stasis for presumably approximately 27 years and woken up presumably approximately 2029/2030.
2003: Half-Life, Portal.
2029/2030: Half-Life 2 (and Episodes), Portal 2.
These dates are estimations using information from the Combine Overwiki. The 2003 info is taken from a December calander in Half-Life (Office Complex) and the date is either 2003 or 2008. The 27 years info is taken from the Overwiki, in-game evidence and fan information.
I believe that it was somewhere around 30,000 years in the future.
A. Paper, wood, upholstery? This is Aperture Science! It would be no surprise to me if they were created to last. They obviously prepared for the apocalypse and extreme periods of time (announcer recordings referring to the future: "If the laws of physics no longer apply in the future then god help you" or "...when society has rebuilt itself").
B. I like to think of the 9,999,999 days in a comical way: Wheatley, being the impressively engineered moron that he is notices that there are an awful lot of nines scrolling on the monitor and realizes what his job originally was. He then decides to wake you (remember that the last time you checked everyone looked pretty much alive).
C. Even for 27 years I doubt that there would be so many plants all over the facility. Despite the super potato growth that little Chell presented on that fateful bring your daughter to work day (which is the perfect time to have her tested).
D. Hundreds of years is always mentioned everywhere, but unless 9,999,999 is a broken overflow timer it would seem unlikely that it was 273 years versus 27,300. In all reality it was probably longer than that.
I think 9 years because the Aperture Labs are near the Great Lakes and way below the water table. Also the labs are nuclear powered and if it was longer, Chell would be dead. Half-Life 2 is also taking place after the time Portal 2 is, so these are my reasons.
I think it is 27 years, as 27 years is enough time for the facility to get into the condition it was in at the begginning, yet if it were any longer, well, I doubt anything would be functional.