We study the dynamic cache dimensioning problem, where the objective is to decide how much storage to place in the cache to minimize the total costs with respect to the storage and content delivery latency. We formulate this problem as a Markov decision process, which turns out to be a restless multi-armed bandit problem and is provably hard to solve. For given dimensioning decisions, it is possible to develop solutions based on the celebrated Whittle index policy. However, Whittle index policy has not been studied for dynamic cache dimensioning, mainly because cache dimensioning needs to be repeatedly solved and jointly optimized with content caching. To overcome this difficulty, we propose a low-complexity fluid Whittle index policy, which jointly determines dimensioning and content caching. We show that this policy is asymptotically optimal. We further develop a lightweight reinforcement learning augmented algorithm dubbed fW-UCB when the content request and delivery rates are unavailable. fW-UCB is shown to achieve a sub-linear regret as it fully exploits the structure of the near-optimal fluid Whittle index policy and hence can be easily implemented. Extensive simulations using real traces support our theoretical results.