Context. Binary stellar systems form a large fraction of the Galaxy’s stars. They are useful as laboratories for studying the physical processes taking place within stars, and must be correctly taken into account when observations of stars are used to study the structure and evolution of the Galaxy. The advent of large-scale spectroscopic and photometric surveys allows us to obtain large samples of binaries that permit characterising their populations. Aims. We aim to obtain a large sample of double-lined spectroscopic binaries (SB2s) by analysis of spectra from the GALAH survey in combination with photometric and astrometric data. A combined analysis will provide stellar parameters of thousands of binary stars that can be combined to form statistical observables of a given population. We aim to produce a catalogue of well-characterised systems, which can in turn be compared to models of populations of binary stars, or to follow-up individual systems of interest. Methods. We obtained a list of candidate SB2 systems from a t-distributed stochastic neighbour embedding (t-SNE) classification and a cross-correlation analysis of GALAH spectra. To compute parameters of the primary and secondary star, we used a Bayesian approach that includes a parallax prior from Gaia DR2, spectra from GALAH, and apparent magnitudes from APASS, Gaia DR2, 2MASS, and WISE. We used a Markov chain Monte Carlo approach to sample the posterior distributions of the following model parameters for the two stars: T_eff[1,2], logg_[1,2], [Fe/H], V_r[1,2], v_mic[1,2], v_broad[1,2], R_[1,2], and E(B−V). Results. We present results for 12 760 binary stars detected as SB2s. We construct the statistical observables T₁∕T₂, ΔV_r, and R₁∕R₂, which demonstrate that our sample mostly consists of dwarfs, with a significant fraction of evolved stars and several dozen members of the giant branch. The majority of these binary stars is concentrated at the lower boundary of the ΔV_r distribution, and the R₁∕R₂ ratio is mostly close to unity. The derived metallicity of our binary stars is statistically lower than that of single dwarf stars from the same magnitude-limited sample. Conclusions. Our sample of binary stars represents a large population of well-characterised double-lined spectroscopic binaries that are appropriate for statistical studies of the binary populations. The derived stellar properties and their distributions show trends that are expected for a population of close binary stars (a < 10 AU) detected through double lines in their spectra. Our detection technique allows us to probe binary systems with mass ratios 0.5 ≤q ≤ 1.