WordNet sense 1 (a particular environment or walk of life):

area, arena, domain, field, orbit

WordNet sense 2 (the geographical area in which one nation is very influential):

sphere of influence

WordNet sense 3 (the apparent surface of the imaginary sphere on which celestial bodies appear to be projected):

empyrean, firmament, heavens, welkin, celestial sphere, vault of heaven

WordNet sense 4 (a particular aspect of life or activity):

sector