Ionized outflows, revealed by broad asymmetric wings of the [OIII] line, are commonly observed in AGN but the low intrinsic spatial resolution of observations has generally prevented a detailed characterization of their properties. The MAGNUM survey aims at overcoming these limitations by focusing on the nearest AGN, including NGC 1365, a nearby Seyfert galaxy (D~17 Mpc), hosting a low-luminosity AGN (Lbol ~ 2x10^43 erg/s). We want to obtain a detailed picture of the ionized gas in the central ~5 kpc of NGC 1365 in terms of physical properties, kinematics, and ionization mechanisms. We also aim to characterize the warm ionized outflow as a function of distance from the nucleus and its relation with the nuclear X-ray wind. We employed VLT/MUSE optical integral field spectroscopic observations to investigate the warm ionized gas and Chandra ACIS-S X-ray data for the hot highly-ionized phase. We obtained flux, kinematic, and diagnostic maps of the optical emission lines, which we used to disentangle outflows from disk motions and measure the gas properties down to a spatial resolution of ~70 pc. [OIII] emission mostly traces an AGN-ionized kpc-scale biconical outflow with velocities up to ~200 km/s. H{\alpha} emission traces instead star formation in a circumnuclear ring and along the bar, where we detect non-circular motions. Soft X-rays are mostly due to thermal emission from the star-forming regions, but we could isolate the AGN photoionized component which matches the [OIII] emission. The mass outflow rate of the extended ionized outflow matches that of the nuclear X-ray wind and then decreases with radius. However, the hard X-ray emission from the circumnuclear ring suggests that star formation might contribute to the outflow. The integrated mass outflow rate, kinetic energy rate, and outflow velocity are broadly consistent with the typical relations observed in more luminous AGN.