This doesn't seem right to me, why don't astronauts suffocate on the International Space Station? You can pressurize air in a spacecraft, I believe the ISS is kept at ~1 atm.
The habitat in Elysium is quite bizarre. The entire donut is built Cabriolet style flying LEO and protagonists just lowers commandeered space helicopters through inner perimeter. It could be said the Chekhov's gun principle applied beautifully but quite bizarre.
Also there's a scene at the beginning where the main character looks up and it's just hanging up there, seemingly stationary. At LEO it should be zapping across the sky (the ISS orbits the Earth every 90 minutes).
OP is referencing an open air habitat so a comparison with the ISS doesn't make sense. Open air designs do actually work once you scale up the size of the habitat enough. This is where we get the idea of Bishop rings which use a wall a hundred or so km in height to keep the air in. Mckendree cylinders (which are a supersized version of O'Neil cylinders) can also have end caps that are open provided you have a high enough wall.
The main advantage of open air designs is it allows you to use aerobreaking when approaching the habitat which could be a significant save in fuel.
The Elysium habitat in the film has (from memory) walls only hundreds of metres high. So there is no way that would contain an atmosphere for any length of time. I guess their could be some sort of high tech field that kept it in, but it would have to be something that doesn't stop a shuttle entering (as they do in the film).
Yeah, that wouldn't work at all but then again, inaccurate science was far from the only problem that movie had imo.
One possible solution is to ionise the air near the walls and then use a magnetic field to contain it. This would not really prevent all air leakage but you don't really have to as long as it's substantially reduced since even the relatively diminutive size of the Elysium habitat would contain a fairly formidable volume of air. There can also be outside magnets that can arrest the momentum of the escaped ionised air enough so it falls back down to Earth. Then all you need is a tether extending down to the atmosphere with an internal air pump and you have a self contained cycle.
The Elysium habitat in the movie is open topped with walls on a spinning torus not a sealed container like the ISS. On Earth this works because of the combination of gravity and the size of the atmosphere pressing on itself.