This is the 4630th most frequent Turkish word.
birilerinin
"Birilerinin" means "someone's" or "some people's" in English.
The word 'birilerinin' is used here to indicate a non-specific group of people belonging to 'birileri' (someone or some people), and in this sentence it's in the genitive case to show possession of the subject (who needs to solve it).
Birilerinin bu sorunu çözmesi gerekiyor.
Someone needs to solve this problem.
In this context, 'birilerinin' again represents an indefinite or unknown person or people, and it is in the genitive case to describe the possession of the 'room' being referred to.
Kütüphane kapanınca birilerinin odasına gittik.
After the library closed, we went to someone's room.
Here, 'birilerinin' refers broadly to an unspecified group who should help, expressing possession or association with the verb 'yardım etmek' (to help).