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Massive stars are often found in multiple systems, yet how binary-star systems with very close separations (<inline-formula><tex-math>$\lesssim$</tex-math></inline-formula> au) assemble remains unresolved. We investigate the formation and inward migration of massive-star binaries in solar-metallicity environments using the star-cluster formation simulation that forms a <inline-formula><tex-math>$1200\, \mathrm{ M}_\odot$</tex-math></inline-formula> stellar cluster and resolves binaries down to 1 au separation. Our results indicate that stars more massive than <inline-formula><tex-math>$2\, {\rm M}_{\odot }$</tex-math></inline-formula> predominantly assemble in binary or triple configurations, in agreement with observations, with member stars forming nearly coevally. In most of these systems, the inner binary hardens by one to three orders of magnitude and reaches a steady-state within the first <inline-formula><tex-math>$0.1\,$</tex-math></inline-formula>Myr. Notably, all binaries whose final separations are below 10 au are hardened with the aid of circumbinary discs, highlighting disc-driven migration as a key to produce tight massive binaries. We further find that binaries form with random inclinations relative to the initial rotation axis of the cloud, and that mutual inclinations in triple systems follow an isotropic distribution, implying that stochastic interactions driven by turbulence and few-body dynamics are crucial during assembly and migration. Finally, stars with <inline-formula><tex-math>$M\gt 2\, {\rm M}_{\odot }$</tex-math></inline-formula> often undergo repeated merger events during cluster evolution, yielding extreme mass ratios (<inline-formula><tex-math>$q\lt 0.1$</tex-math></inline-formula>). Some of these products may evolve into compact-object binaries containing a black hole or neutron star, including X-ray binaries and systems detectable by Gaia.