Departure time choice models play a crucial role in determining the traffic load in transportation systems. Most studies that consider departure time user equilibrium (DTUE) problems make assumptions on the user characteristics (e.g., distribution of desired arrival time and trip length) or dynamic traffic model (e.g., classic bathtub or point queue models) in order to analyze the problem. This paper relaxes these assumptions and introduces a new framework to model and analyze the DTUE problem based on the so-called mean field games (MFGs) theory. MFGs allow us to define players at the microscopic level similar to classical game theory models, translating the effect of players' decisions to macroscopic models. In this paper, we first present a continuous departure time choice model and investigate the equilibria of the system. Specifically, we demonstrate the existence of the equilibrium and characterize the DTUE. Then, a discrete approximation of the system is provided based on deterministic differential game models to numerically obtain the equilibrium of the system. To examine the efficiency of the proposed model, we compare it with the departure time choice models in the literature. We apply our framework to a standard test case and observe that the solutions obtained based on our model are 5.6% better in terms of relative cost compared with the solutions determined based on previous studies. Moreover, our proposed model converges with fewer iterations than the reference solution method in the literature. Finally, the model is scaled up to the real test case corresponding to the whole Lyon metropolis with a real demand pattern. The results show that the proposed framework is able to tackle a much larger test case than usual to include multiple preferred travel times and heterogeneous trip lengths more accurately than existing models.