We demonstrate short-range terahertz radar based on a leaky-wave antenna with a beam steering capability. As a proof of concept, we develop a microstrip-based periodic leaky-wave antenna driven by a vector network analyzer. By sweeping the frequency from 235 to 325 GHz, beam steering from -23° to +15° across the broadside can be achieved with a nearly constant beam width of 4°. Small target detection is demonstrated by locating a metal cylinder with a diameter of 12 mm placed 46-86 mm in front of the antenna with a mean error of 2.4 mm. The use of a leaky-wave antenna can pave the way for developing a low-loss, low-profile, and wide-aperture terahertz radar. Importantly, it can be integrated with a solid-state source and a detector. The proposed approach is particularly promising for use with emerging small devices such as drones or wearable devices, where millimeter-wave radar is not suitable in terms of the resolution and system footprint.