Surveys in the Milky Way and Large Magellanic Cloud have revealed that the majority of massive stars will interact with companions during their lives. However, knowledge of the binary properties of massive stars at low metallicity, and therefore in conditions approaching those of the Early Universe, remain sparse. We present the Binarity at LOw Metallicity (BLOeM) campaign, an ESO large programme designed to obtain 25 epochs of spectroscopy for 929 massive stars in the Small Magellanic Cloud, allowing us to probe multiplicity in the lowest-metallicity conditions to date (Z=0/2Z_{sun}_). BLOeM will provide (i) the binary fraction, (ii) the orbital configurations of systems with periods of P<~3yr, (iii) dormant black-hole binary candidates (OB+BH), and (iv) a legacy database of physical parameters of massive stars at low metallicity. Main sequence (OB-type) and evolved (OBAF-type) massive stars are observed with the LR02 setup of the giraffe instrument of the Very Large Telescope (3960-4570{AA} resolving power R=6200; typical signal-to-noise ratio (S/N)~70-100). This paper utilises the first nine epochs obtained over a three-month time period. We describe the survey and data reduction, perform a spectral classification of the stacked spectra, and construct a Hertzsprung-Russell diagram of the sample via spectral-type and photometric calibrations. Our detailed classification reveals that the sample covers spectral types from O4 to F5, spanning the effective temperature and luminosity ranges 6:5<~Te/kK<~45 and 3.7<logL/L_{sun}_<6.1 and initial masses of 8<~M_ini_<~80M_{sun}_. The sample comprises 159 O-type stars, 331 early B-type (B0-3) dwarfs and giants (luminosity classes V-III), 303 early B-type supergiants (II-I), and 136 late-type BAF supergiants. At least 82 stars are OBe stars: 20 O-type and 62 B-type (13% and 11% of the respective samples). In addition, the sample includes 4 high-mass X-ray binaries, 3 stars resembling luminous blue variables, 2 bloated stripped-star candidates, 2 candidate magnetic stars, and 74 eclipsing binaries.