Janus MoSSe monolayers were recently synthesised by replacing S by Se on one side of MoS$_2$ (or vice versa for MoSe$_2$). Due to the different electronegativity of S and Se these structures carry a finite out-of-plane dipole moment. As we show here by means of density functional theory (DFT) calculations, this intrinsic dipole leads to the formation of built-in electric fields when the monolayers are stacked to form $N$-layer structures. For sufficiently thin structures ($N<4$) the dipoles add up and shift the vacuum level on the two sides of the film by $\sim N \cdot 0.7$ eV. However, for thicker films charge transfer occurs between the outermost layers forming atomically thin n- and p-doped electron gasses at the two surfaces. The doping concentration can be tuned between about $5\cdot 10^{12}$ e/cm$^{2}$ and $2\cdot 10^{13}$ e/cm$^{2}$ by varying the film thickness. The surface charges counteract the static dipoles leading to saturation of the vacuum level shift at around 2.2 eV for $N>4$. Based on band structure calculations and the Mott-Wannier exciton model, we compute the energies of intra- and interlayer excitons as a function of film thickness suggesting that the Janus multilayer films are ideally suited for achieving ultrafast charge separation over atomic length scales without chemical doping or applied electric fields. Finally, we explore a number of other potentially synthesisable 2D Janus structures with different band gaps and internal dipole moments. Our results open new opportunities for ultrathin opto-electronic components such as tunnel diodes, photo-detectors, or solar cells.