In Ruby you have to use square brackets to access hashes.
json =JSON.parse(File.read('input.json'))
domains = []
json.Results.map{|result| result.Paths.map{|path| domains << path.Domain }}
However it is Ruby... so you also can override the Hash class and access your Hashes with a simple dot notation. (by the simple solution of: @papirtiger )
For example: domain = json.Results.Paths.Domain
require 'ostruct'
JSON.parse(File.read('input.json'), object_class: OpenStruct)