Microwave imaging (MWI) systems are usually enclosed within casings, e.g., in order to contain the utilized coupling liquid or to help mount the antenna system. On the other hand, inverse scattering algorithms, which are used to process the measured microwave scattering data, often assume that the background medium of the imaging system extends to infinity (i.e., unbounded background medium assumption). Thus, they do not consider the reflections occurring at the system enclosure. For such algorithms to yield successful images, these reflections need to be minimized, e.g., via the use of a lossy coupling liquid. As an alternative to a lossy background medium which also reduces the desired signal level, this paper investigates the use of metallic-backed absorbing metasurfaces as the MWI system enclosure in order to (i) reduce these reflections, and also (ii) to shield the MWI system from external interference. Using simulated data, we then show that standard inverse scattering algorithms, employing the free-space assumption, can successfully process the data collected under the metasurface enclosure and yield acceptable permittivity images. The advantages and disadvantages of absorbing metasurface enclosure, along with the limitations of this study, will also be discussed. Finally, an absorbing metasurface is fabricated and its reflectivity is experimentally evaluated.