Why capital ships separated guns and machinery

What cooling technology sat between battleship turrets and engines?

Usually not a special cooling system: distance, compartments, insulation, and intervening spaces did most of the work.

Battleships generally relied on physical separation instead of a special cooling barrier between magazines and machinery. Water tanks, occupied compartments, insulated bulkheads, and other intervening spaces provided distance and isolated heat. In the Nelson design family, the superstructure and its supporting systems occupied the central zone while the main armament and magazines remained forward of the machinery. Refrigeration equipment might appear in the intervening area, but it was not the protective principle because refrigeration plants also produce heat. Arrangement, insulation, and damage-control zoning worked together in capital-ship design.