Q:

What does "fabric of his existence" means?

I heard this sentence in movie "The Fisher King" 

See, guy goes to work every day, eight hours a day, seven days a week.
  Gets his nuts so tight in a vice, he starts questioning the very fabric
  of his existence.

A:

Your phrase 

fabric of his existence

has the metaphorical meaning that it is the basis or foundation of his existence.
A piece of fabric has many interwoven threads and if something is

part of the fabric

it means it is an integral part of the material or the whole.

