If I make a good backstory for my character is it possible to get an animal companion or two due to my backstory if the game master accepts it? The animals I want are an owlbear and a chimaera. I want to have them now.
That is entirely up to your GM
D&D, as with most RPGs, has "rule 0", which basically states the GM/DM gets the final say in anything.
There is no existing rule in D&D that allows you to get animal companions via backstory. On top of that, the two creatures you have listed are monstrosities, not beasts as animal companions typically are.
If you are starting a new low-level character, I very much doubt most GMs would allow it, since those two creatures would easily be stronger than all members of a low-level party combined.