If you’re in Mecca which way do you face when you pray?
You face the ka’aba.
I’d imagine towards the Kaaba. Where else would they face?
Yeah, but what if you’re standing at the exact center of the Kaaba? 
nevermind.
There’s sort of a giant rock there - you can’t quite stand in the middle of a rock.
The Kabba is a big cube that is rarely opened so nobody goes inside. If you are in the courtyard of the mosque, you still face the Kabba.
If you’re in there, you already know which way to face…
It’s hollow, and there is a ritual cleaning of it every so often. I do wonder what that involves.
Just ask the guy praying here.
The rock inside the Ka’aba is a big meteorite, right?
Digging around on wiki, it looks like the Kaaba is a fractured rock about the size of a basketball. But yeah - I’m guessing meteorite. (And I guess a basketball-sized meteorite qualifies as “big”, doesn’t it?) It also appears that it’s not inside the shrine, but mounted into one of the corners. I bet the bigwigs just set up a big screen TV inside the shrine and watch football all day. (Because that’s what I’d do.)
A picture of the Black Stone. It is in the center of this silver encasement.
http://www.google.com/imgres?imgurl=http://2.bp.blogspot.com/_TuseS_84DVI/TOFawSfA3gI/AAAAAAAAAj8/KSDi_jcIXMY/s1600/kaaba-3-large.jpg&imgrefurl=http://helinamin.blogspot.com/2010/11/according-to-muslims-four-thousand.html&h=432&w=360&sz=39&tbnid=UXW5vyevpsVooM:&tbnh=126&tbnw=105&prev=/images%3Fq%3Dkaaba%2Bblack%2Bstone&zoom=1&q=kaaba+black+stone&hl=en&usg=__ooreKuAw60r-CF-ZWHXctBk2lKs=&sa=X&ei=2lQjTdStKZCusAPbhaT3Cg&sqi=2&ved=0CCsQ9QEwAg
Heck the Cube is opened a couple of times a year. I personally know people who have been in it. It is no secret or anything.
The King opens it once a year to ceremonially clean it. He is followed by a gaggle of ambassadors and then a line of the general public.
While in the Cube, one may pray in any direction. Needless to say, people visiting take advantage of this fact and make a quick bow toward each of the four corners and then leave to brag about their accomplishment for years to come.
Actually, the subject of qibla is nothing to sneeze in the direction of.
If you’re 50 miles away, it’s pretty simple.
But if you’re thousands of miles away, do you face Mecca directly through the earth? Or across a great circle route?
And Allah help you with the complications of being a Muslim astronaut (in a near-earth orbit, as most manned space flight is these days). Imagine rotating while kneeling.
As others have noted, the Black Stone is in a silver frame in one corner of the Ka’aba. The magazine Popular Astronomy ran a lengthy article on the Stone back in the 1930s, I think (and without reaching a conclusion about the Stone’s true nature, IIRC). In the 1970s, the magazine Meteorics ran a short article by a Muslim astronomer who made a close examination of the Stone while on the hajj. His conclusion was that it was not a meteorite (the article was anonymous – no need to get in trouble with people who wouldn’t appreciate the mixture of faith and science).
Regarding the qibla, I think a more interesting question is what direction you face when you’re on the point on the Earth diametrically opposite to Mecca (you’d have to be on a ship – it’s in the Pacific Ocean). Does it make a difference which way you face? Or does the slight difference in around-the-earth distance because of the non-sphericity of the earth make enough of a difference to give a preferred direction.
If you’re in Mecca, as noted, you face the Ka’aba. There are plenty of photos of the Square during the hajj showing the pilgrims forming concentric circles around the shrine.
Mecca can be oddly disorienting to a visitor. In every other Saudi city the mosques all point the same way, but of course in Mecca they point Cube-ward.
The Black Stone is most likely a meteorite. That is how it is commonly described. When the first version of the Cube was built, the builders found a neat stone and said, ‘How neat,’ and included it. It is not really sacred or anything.
Shouldn’t really be a problem. Like with astronauts, you just give it your best guess and be sincere in your prayers. Islam is very forgiving that way. It’s not like missing the proper direction by a few degrees will invalidate your prayers or anything. As long as you make a sincere attempt, you’re golden.
Yeah, Allah has a UZR of 1.0.
I’ve wondered what will happen after, say, Mars is colonized. I figure they’ll just set up a local equivalent and point toward that.