without getting into the complexity of a foreign own corporation (then you can buy as many as you want or can), by Law in China foreigners can buy one apartment anywhere in China they are allowed to visit of live, with the following conditions.
1.- At the time of title transfer, you must have resided in China at least one year.  This is where the temporary residence permits come in handy.
2.- You must use the apartment to live on yourself, can not be rented out or for other business purposes.
3.- And of course, it goes without saying, must have the money to pay for it in full, and if new the money to pay for finishing the inside and furnishing it.  It is next to impossible with current market conditions to get a bank loan if you are a foreigner.  Not even with a 50 % down payment.  Now, if you can get private finnancing, that is allowed.

Not everyone. Beijingers and someone who lived in Beijing for a while

why would you want to buy property here? no one owns property here... they dont even own the space, they just own the rights...

It depends on what part of Beijing.  If it is within the first 3 rings, you need to be Han Chinese and have extensive family roots in Beijing.

Outside the third ring is open to anyone with the money.

Yes anybody can...there is a place that use to be only for westerners and now they made it so everybody in Beijing can own the property...so its everyone..