I will have two transit flights in China within a month (31st Dec and 21st Jan) - can I get 2x 24h f

If your layovers in Beijing are shorter than 24 hours, you will be covered by 24 hour visa free transit and you do not need any kind of visa. However, if your stay will be longer than 24 hours, you will be eligible for 144 hour visa free transit.

Now, to be allowed to exit the airport is another thing. When traveling on 24 hour visa free transit, a passenger has a right to travel visa free and to stay airside while waiting for the onward/connecting flight. However, if a passenger under 24 hour visa free transit wants to exit the airport she/he will apply for a temporary entry permit at 24/144 hour visa free transit immediately upon landing. This application is for free and decision whether the permit will be granted or not depends entirely on discretionary decision of the airport authorities on a day. If your layover in Beijing is less than 8 hours long you should not even try to apply because you will be rejected, but if your layover is longer, especially if it is during the night time, then you will book an accommodation and present it to the authorities as a printout. Also, booking a tour in Beijing also helps.

In other words, you can always apply for a temporary entry permit, but you have to be ready to accept both outcomes - positive or negative one... I hope you will receive positive outcomes.