At this time, any astronauts or cosmonauts who run into trouble while in space are largely on their own. If the ISS suffers from a failure that prevents the inhabitants from evacuating, or someone gets sufficiently injured while on a mission, or anything else that puts the crew at risk without the ability to deal with it themselves, people may die and the future of space exploration will be in jeopardy. That's where a Space Rescue Corp comes in.
The Space Rescue Corp (SRC), as I imagine it, would have a team of three, at least one of which would be an MD. These three would be ready to go within 24 hours of an emergency being declared. As the cost of placing people into orbit declines and the reliability of the equipment being used increases, the SRC makes more and more sense, though a few details immediately stand out.
Training. The Corp would need to be trained in both medical and engineering matter in order to effect a rescue. As mentioned, at least one of the crew would need to be an MD, preferably one who can do surgery. The other two would need to have detailed medical training, mostly to assist the doctor, but also to still be able to help if the doctor is incapacitated. The entire crew would need to know how to handle the equipment required to be able to get to those who require help. This could be a malfunctioning airlock, suit, or any other type of equipment. I believe that much of this training is already used.
Equipment. Every vehicle or station that goes up would need to be accessible from the outside, in case none of those inside were able to assist. Any airlock would have to be controllable from outside. The SRC would need to be able to quickly get medical and other rescue equipment into the vehicle/station. I also believe that a lot of this technology is already being used or is at least available for use.
Response. This is the tricky one. There would need to be multiple launch sites to negate the effect of weather on a launch. High winds? Go to a different launch site. Each site would need to have the SRC vehicle ready to go at a moment's notice. This is obviously the sticking point. As we have never truly needed such a group in the past, and considering the high costs of maintaining such the SRC, is it truly worth it? Maybe not now, but I predict that, eventually, we will. It will likely take a few tragic accidents (nobody likes to spend money they don't think they need to), but it will be done. And, eventually, I can easily foresee a permanent location in space itself from which to base such a group. That's likely a hundred years in the future, especially considering the glacial pace of current space exploration, but it will happen.
Off mission duties. Between emergencies, the Corp would need to keep busy and useful. As such, their first and foremost duty would be to expand their training. All Corp members would continually push their medical and engineering knowledge as far as they could. At the same time, they would train other astronauts and cosmonauts to be able to either prevent such emergencies, or to be able to handle them themselves, much as they do now. It would only be during times when, as I said, a crew could not save themselves would the SRC be called in.
I expect to not see such a group in my lifetime. The only way it could exist in the next fifty years is if humanity picked up the slack and took space exploration more seriously. If it does exist, I'll probably be too old to care at that point. I'll be too busy looking for my walker and my teeth.