Today, in technologically advanced countries there are a number of scientific and technical projects to create robotic systems (RS) in various fields of medicine. The additional efforts are wanted for their practical introduction in the extreme environment, in particular, for the rescue of persons of hazardous occupations, which are most at risk when performing their professional activities. In future lunar missions, the particular attention will be necessary for assistance a crew during extravehicular activity (EVA) on the moon's surface. EVA is one of the most important types of flight's operations and requires strict fulfilling the safety requirements, when fulfilling rescue operations in emergency evacuation with using of Autonomous Mobile Robots (AMR). To select potentially suitable solutions to adapt existing land-based rescue robots to the conditions of human exploration of the Moon, it is necessary to turn an attention to the a couple of important characteristics, such as: the weight and size, the load capacity, resistance to overturning when carrying loads comparable in mass with the mass of the robot, off-road capability across rough terrain with various densities of soil, the natural "Human Robot Interaction" on base of multimodal interfaces for remote robot control, and others. The paper gives the general representation of the problem situations on the lunar surface, when a crew in autonomy conditions has great difficulties, that are connected with the rescue and evacuation of a cosmonaut during EVA into the Lunar Lander in case of crashing the spacesuit's life support system and/or the loss of performance by a cosmonaut.