We present a homogeneously selected sample of 15779 candidate binary systems with main sequence primary stars and orbital periods shorter than 5d. The targets were selected from TESS full-frame image light curves on the basis of their tidally induced ellipsoidal modulation. Spectroscopic follow-up suggests a sample purity of 83+/-13 per cent. Injection-recovery tests allow us to estimate our overall completeness as 28+/-3 per cent with P_orb_<3d and to quantify our selection effects. 39+/-4 per cent of our sample are contact binary systems, and we disentangle the period distributions of the contact and detached binaries. We derive the orbital period distribution of the main-sequence binary population at short orbital periods, finding a distribution continuous with the lognormal distribution previously found for solar-type stars at longer periods, but with a significant steepening at P_orb_<~3d, and a pile-up of contact binaries at P_orb_~=0.4d. Companions in the period range of 1-5 dare an order of magnitude more frequent around stars hotter than ~6250K (the Kraft break) when compared to cooler stars, suggesting that magnetic braking shortens the lifetime of cooler binary systems. However, the period distribution in the range 1-10d is independent of temperature. We detect resolved tertiary companions to 9.0+/-0.2 per cent of our binaries with a median separation of 3200au. The frequency of tertiary companions rises to 29+/-5 per cent among the systems with the shortest ellipsoidal periods. This large binary sample with quantified selection effects will be a powerful resource for future studies of detached and contact binary systems with P_orb_<5d.