Is it possible for a place to exist with zero Higgs field, by kind of shielding it or something(theoretically)?
The simple answer is no, though as always the answer is more involved than this.
I would guess you are thinking of the way that a Faraday cage can block electromagnetic fields. However the Higgs field is a quantum field and this is very different from an electromagnetic field. A quantum field is a mathematical object that is present everywhere in spacetime and describes how fundamental particles behave. Quantum fields cannot be screened out because they are not a physical thing.