Context. Ionized outflows, revealed by broad asymmetric wings of the [O III] λ5007 line, are commonly observed in active galactic nuclei (AGN) but the low intrinsic spatial resolution of the 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 active nucleus (Lbol ∼ 2 × 1043 erg s−1). Aims. 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. Methods. We employed optical integral-field spectroscopic observations from VLT/MUSE 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 gravitational motions in the disk and measure the gas properties down to a spatial resolution of ∼70 pc. We then performed imaging spectroscopy on Chandra ACIS-S data guided by the matching with MUSE maps. Results. The [O III] emission mostly traces a kpc-scale biconical outflow ionized by the AGN having velocities up to ∼200 km s−1. Hα emission traces instead star formation in a circumnuclear ring and along the bar, where we detect non-circular streaming gas motions. Soft X-rays are predominantly due to thermal emission from the star-forming regions, but we manage to isolate the AGN photoionized component which nicely matches the [O III] emission. The mass outflow rate of the extended ionized outflow is similar to that of the nuclear X-ray wind and then decreases with radius, implying that the outflow either slows down or that the AGN activity has recently increased. However, the hard X-ray emission from the circumnuclear ring suggests that star formation might in principle 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.