Terahertz radars based on leaky-wave antennas are promising for the realization of radar systems with high resolution over short ranges. This type of radar relies on spatial frequency mapping to realize a wide field of view without mechanical actuation or electrical beam steering. Previously, integrated leaky-wave antennas based on metallic wave confinement have been implemented, but the high ohmic losses limit the realized antenna gain, which is essential for extending the range of such a system when limited power is available. Here, we demonstrate an all-dielectric leaky-wave antenna fabricated monolithically from silicon and then apply leaky-wave radar techniques to realize a terahertz radar system capable of real-time data acquisition. Through this all-dielectric approach, we can avoid metallic losses, achieving an experimentally measured maximum realized gain of 25 dBi and a 34.3° 3-dB field-of-view while utilizing established and scalable fabrication techniques essential for the wide-spread adoption of terahertz technologies. We foresee this technique being applied to a variety of real-time radar applications, and here we demonstrate two potential use-cases: multi-object tracking and differentiation of liquids.