We study the joint operation and sizing of cooling infrastructure for commercial HVAC systems using reinforcement learning, with the objective of minimizing life-cycle cost over a 30-year horizon. The cooling system consists of a fixed-capacity electric chiller and a thermal energy storage (TES) unit, jointly operated to meet stochastic hourly cooling demands under time-varying electricity prices. The life-cycle cost accounts for both capital expenditure and discounted operating cost, including electricity consumption and maintenance. A key challenge arises from the strong asymmetry in capital costs: increasing chiller capacity by one unit is far more expensive than an equivalent increase in TES capacity. As a result, identifying the right combination of chiller and TES sizes, while ensuring zero loss-of-cooling-load under optimal operation, is a non-trivial co-design problem. To address this, we formulate the chiller operation problem for a fixed infrastructure configuration as a finite-horizon Markov Decision Process (MDP), in which the control action is the chiller part-load ratio (PLR). The MDP is solved using a Deep Q Network (DQN) with a constrained action space. The learned DQN RL policy minimizes electricity cost over historical traces of cooling demand and electricity prices. For each candidate chiller-TES sizing configuration, the trained policy is evaluated. We then restrict attention to configurations that fully satisfy the cooling demand and perform a life-cycle cost minimization over this feasible set to identify the cost-optimal infrastructure design. Using this approach, we determine the optimal chiller and thermal energy storage capacities to be 700 and 1500, respectively.