We show that it is impossible to perform ideal projective measurements on quantum systems using finite resources. We identify three fundamental features of an ideal projective measurement and show that when limited by finite resources only one of these features can be salvaged. Using an explicit model of an N-particle detector perfectly reproducing the statistics of the system, we provide tight analytic expressions for the energy cost of performing a measurement. This cost may be broken down into two parts. First, the cost of preparing the pointer in a suitable state, and second, the cost of a global interaction between the system and pointer in order to correlate them. Our results show that, even under the assumption that the interaction can be controlled perfectly, achieving perfect correlation is infinitely expensive. We provide protocols for achieving optimal correlation given finite resources.