We discuss a novel method to address the issue of nuclear safety of the Shelter Object in the New Safe Containment at Chernobyl Nuclear Power Plant, using ultra-sensitive radioxenon measurements to monitor the fission activity. The automatic radioxenon measurement system SAUNA QB, originally developed for nuclear explosion monitoring, could be well suited for the proposed method, and theoretical modelling indicates that the radioxenon activity concentration resulting from fission in the fuel containing material might be at detectable levels inside the New Safe Containment. A simplified sampling and measurement method was developed in order to get a first indication of the, still unknown, radioxenon activity concentration levels inside the shelter. Using this system, air was collected close to fuel containing material in the Shelter Object, and the activity of the collected air was measured. No radioxenon could be detected, but the result indicates an upper activity concentration limit of about one Bq/m3, which is inside the dynamic range for the SAUNA QB. Additional measurements are desirable to confirm the result. To further, and more thoroughly, investigate the feasibility of the proposed method, it is necessary to measure for a longer time period using more sensitive equipment, such as the SAUNA QB.