Reber was born and raised in Chicago, and graduated from the Illinois Institute of Technology in 1933 with a degree in radio engineering. He was a ham radio operator, and worked for various radio manufacturers in Chicago from 1933 to 1947. When he learned of Jansky's work in 1933, he decided this was the field he wanted to work in, and applied to Bell Labs where Jansky was no working. However this was during the height of the Great Depression and there were no jobs available.
Instead Reber decided to build his own radio telescope in his back yard in Wheaton, a suburb of Chicago. His design was considerably more advanced than Jansky's, consisting of a parabolic sheet metal mirror 9 meters in diameter, focusing to a radio receiver 8 meters above the mirror. The entire assembly was mounted on a tilting stand allowing it to be pointed in various directions, although not turned. The telescope was completed in 1937.
Reber's first receiver operated at 3300 MHz and failed to detect signals from outer space, as did his second, operating at 900 MHz. Finally his third attempt at 160 MHz was successful in 1938, confirming Jansky's discovery. Reber turned his attention to making a radio-frequency sky map, which he completed in 1941 and extended in 1943. He published a considerable body of work during this era, and was largely responsible for the "explosion" of radio astronomy in the immediate post-WWII era.
During this time he uncovered a mystery that was not explained until the 1950s. The standard theory of radio emmissions from space was that they were due to blackbody radiation, light (of which radio is a non-visible form) that is given off by all hot bodies. Using this theory one would expect that there would be considerably more high-energy light than low-energy due to the presense of stars and other hot bodies. However Reber demonstrated that the reverse was true, and that there was a considerable amount of low-energy radio signal. It was not until the 1950s that synchrotron radiation was offered as an explaination for these measurements.
Reber later donated his telescope to the NRAO in Green Bank, West Virginia, and helped supervise it's re-construction at that site. The telescope was then mounted on a turntable, allowing it to be pointed in any direction. Reber also helped with a reconstruction of Jansky's original telescope as well. In all Reber spent four years working for the National Bureau of Standards.
In the 1950s he wanted to return to active studies but much of the field was already filled with very large and expensive instruments. Instead he turned to a field that was being largely ignored, that of very low-frequency radio signals. However most low-frequency signals are filtered out by the Earth's atmosphere, so Reber moved to Tasmania to start his observations there. He died in Tasmania in 2002.