One idea could be that the Avengers themselves get injured during a major battle and are taken to a special hospital. There, they have to deal with not only their physical wounds but also the psychological trauma. For example, Iron Man might struggle with the fear of his suit failing again while in recovery. The doctors and nurses in the hospital could be normal humans who are in awe of the Avengers but also have to keep their professional composure.
Maybe the hospital is secretly run by a villain who is using the Avengers' injuries as an opportunity to study their powers. The Avengers start to notice strange things happening around the hospital, like their powers being drained slightly. Hawkeye could be the first to suspect something is wrong as he is very observant. They then have to figure out what's going on and escape the hospital while still injured.
In the fanfiction, the Avengers could be visiting a hospital to see a young fan who is terminally ill. They end up spending time with other sick children as well. Captain America could inspire the kids with his stories of courage. This visit could also lead to the Avengers funding a new wing in the hospital dedicated to treating superhero - related injuries that normal hospitals are not equipped to handle.