Aims: Highly resolved maps of the local Galactic dust are an important ingredient for sky emission models. In nearly the whole electromagnetic spectrum one can see imprints of dust, many of which originate from dust clouds within 300pc. Having a detailed 3D reconstruction of these local dust clouds enables detailed studies, helps to quantify the impact on other observables and is a milestone necessary to enable larger reconstructions, as every sightline for more distant objects will pass through the local dust. Methods: To infer the dust density we use parallax and absorption estimates published by the Gaia collaboration in their second data release. We model the dust as a log-normal process using a hierarchical Bayesian model. We also infer non-parametrically the kernel of the log-normal process, which corresponds to the physical spatial correlation power spectrum of the log-density. Results: Using only Gaia data of the second Gaia data release, we reconstruct the 3D dust density and its spatial correlation spectrum in a 600pc cube centered on the Sun. We report a spectral index of the logarithmic dust density of $3.1$ on Fourier scales with wavelengths between 2pc and 125pc. The resulting 3D dust map as well as the power spectrum and posterior samples are publicly available for download.