The strong mass loss of Luminous Blue Variables (LBVs) is thought to play a critical role in massive-star evolution, but their place in the evolutionary sequence remains debated. A key to understanding their peculiar instability is their high observed luminosities, which often depends on uncertain distances. Here we report direct distances and space motions of four canonical Milky Way LBVs---AG~Car, HR~Car, HD~168607, and (candidate) Hen~3-519---from the Gaia first data release. Whereas the distances of HR~Car and HD~168607 are consistent with previous literature estimates within the considerable uncertainties, Hen~3-519 and AG~Car, both at $sim$2~kpc, are much closer than the 6--8~kpc distances previously assumed. As a result, Hen~3-519 moves far from the locus of LBVs on the HR Diagram, making it a much less luminous object. For AG~Car, considered a defining example of a classical LBV, its lower luminosity would also move it off the S~Dor instability strip. Lower luminosities allow both AG~Car and Hen~3-519 to have passed through a previous red supergiant phase, lower the mass estimates for their shell nebulae, and imply that binary evolution is needed to account for their peculiarities. These results may also impact our understanding of LBVs as potential supernova progenitors and their isolated environments. Improved distances will be provided in the Gaia second data release, which will include additional LBVs. AG~Car and Hen~3-519 hint that this new information may alter our traditional view of LBVs.
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