Spoilers here--but it's not really a movie that will be spoiled by hearing some of the plot.
Though I'll start with a confession: I did not love District 9. I respected it a lot for some of the choices it made; I think there's a lot in it to enjoy and discuss. Particularly the way in which it took the Dances With Wolves / Last Samurai / Avatar conceit that a white guy is always better at being the natives than the natives are and twisted it: so Wikus ends up being in an alien body, but he's kind of a terrible mess for most of the movie; and most importantly, he doesn't want to be in that body. Sure, he's got some cool new powers, but it comes with a pretty serious cost. That said, the second half of the movie seemed to devolve into pure shoot-em-up video game cliches that left me bored. And I like video games!
Elysium follows the same basic formula, in some ways: there's a guy who just wants to get on with his life, which is being an ex-con who is still in love with his childhood sweetheart, and who works a shitty factory job. It's a pretty simple character--and I mean that in a good way. He is not a guy who is interested in fighting the system, he just wants love. Nice and simple--and not at all necessary to show all the flashbacks to his childhood, which don't add much. It also doesn't help that his childhood love has left him and, when they meet again, shows no particular interest in him. I don't mind a protagonist who is selfish, in some ways, but it's a little hard to get on the side of a guy who seems mostly interested in a woman who's not interested in him.
And just as Wikus in District 9 gets catapulted into action through an accident, our protagonist here is pushed into action by an industrial accident, which nicely fits with the theme of economic exploitation, but doesn't give him a lot of agency. And then when he's given some super-powers through an exoskeleton, that sort of seals his fate as the Wikus of this film: a guy who likes the status quo (though here, on the bottom of things), who gets pushed into action through an accident, and who gets special powers that help him.
So I've hinted at some reasons that character doesn't really carry the film: not a lot of agency; some wish-fulfillment for powers (that doesn't really actually fit the movie's theme or plot and could easily be done away with); and a goal that is a little off-putting (to make this woman love him).
Sometimes a movie can skate by with a flawed protagonist if the secondary characters are interesting. Unfortunately, that's not really the case here. The hacker who helps the protagonist starts out as a greedy black-market figure, but--for no reason that I can see--morphs into a crusader for justice. The evil bureaucrat and the evil corporate figure and the evil soldier are all so patently evil that it's uninteresting to watch them as characters; they become mere features of the landscape, obstacles for the protagonist to overcome. As in District 9's devolution to shoot-em-up, there's very little here beyond the pleasure of watching the good guy shoot up the bad guys.
To be clear, none of these problems sinks the film utterly; each just saps a little bit of vitality from it.
And there is one, final, big problem to the film, thematically: it's all about the haves vs. the have-nots, where the thing that people need most is medical attention. I can dig that as a set-up. But as others have pointed out, there's no issue of scarcity or medically-related oppression. The rich keep the medicine for themselves because they just do. And (spoiler) once all of the medicine gets distributed, everyone is healthy, with no issues of scarcity or cost coming up. In another type of movie, this would be fine; this would be part of the admission price: "If you want to watch this film, you have to accept that the space condo people are hoarding the medicine for themselves." It's the sort of "buy-in" or "gimme" that lots of films and tv shows start with: you have to believe in X--a doctor with an amazing diagnostic mind (for House), that New York City is incredibly crime ridden (for all those Law & Orders), or whatever it is.
But Elysium is about that situation. So for us to take it seriously, it needs to show us more of the whys and wherefores. Otherwise, it's not about us or people like us, but about cartoon villains who for no reason are killing the poor inhabitants of Earth. And then it becomes just a film where a good guy who just wanted to live a normal, poor life, goes out and righteously kills rich people.