Funny scary story

One night, I was staying at school to work on some mathematics when I decided to go home.
While I was walking through the school, I heard someone nearby.
I was afraid because it was quite late and there were not a lot people out therearound.
Besides, I had to walk behind some containers to reach a crossroad and then find the school’s door.
I was arriving at the crossroad when I saw the people that I heard before. She seemed to come straight at me so I became more scared. I was staring at her and started following another path when it turned out that she had not even noticed my presence and she was just following her own way.

Note 1: I've left the first sentence mostly as it is but the sentence structure is a little unusual so it would be better to re-word completely. Perhaps to something like: "One night, after staying at school to work on some mathematics for several hours, I decided to go home."

Note 2: 'Someone' is singular, 'the people' is plural and 'she' is singular but it sounds like you are trying to refer to the same person throughout. You also shouldn't say 'she' without first making some mention of the woman you are referring to (both 'someone' and 'the people' are gender-neutral).  

Funny scary story

One night, I was staying at school to work on mathematics when I decided to go home.
While I was walking through the school, I heard someone nearBY..
I [got afraid] because it was quite late, and there was not a lot people out there.
Besides, I had to walk behind some containers to reach a crossroad and then find the school’s door.
[I was arriving in the crossroad] when I saw the people that I HAD heard before. [She] [THIS WOMAN] seemed to come straight TOWARD me so I got more scared. I was staring at her and starting TO FOLLOW another path when it turned out that she had not even noticed my presence and she was just [following] her own way.

[got afraid] more common would be I GOT SCARED
[I was arriving in the crossroad] UNDERSTANDABLE BUT AWKWARD. I WOULD HAVE SAID [I was at an intersection...]

[She] YOU DID NOT INTRODUCE HER BEFORE SO THE READER HAS NO IDEA WHO THIS IS

[following] THIS IS OK BUT SOUNDS A BIT NON NATIVE--I'D SAY SOMETHING LIKE [SHE WAS JUST GOING ABOUT HER DAY]