"I work as Paramedic in a busy 911 system so I can give you my medical opinion.
The chances of Joel surviving his fall and not recieving immediate medical assistance and I mean immediate is slim to none. There are crazy circumstances were people do survive those things. I have seen someone get shot in the head and survive somehow, so it is not impossible but incredibly unlikely. When Joel was impaled in the abdomen you assume he did not hit any large organs or anything vital. But beyond that your talking about big time blood loss, he had no time to attempt to stop the bleeding because he was fighting enemies. You also run the risk of that rod he was impaled on going through your spine and paralyzing you if not killing you. Then you worry about infection, sure Ellie got him antibiotics but the amount she got would not come close to resolving that infection.
He fell on a rusty, filthy rod and had an open abdominal wound for who knows how long. He would need very strong IV antibiotics, I stress the word IV and a tons of fluids and a long hospital stay to recover and thats on the off chance that he would survive to begin with. So your answer from me is the chances of him surviving the initial fall is extremely slim, surviving without getting immediate medical attention, nearly impossible. Surviving the infection from the rod without very powerful antibiotics, impossible. So as you see, survival for Joel would be a medical miracle."
That whole sequence of events disturbed me more than anything else. My suspension of disbelief broke at that point. Other than that though, story's pretty good and it has one of the most powerful opening sequences I've ever seen (although a bit contrived). Really pulled me in.