Basically when Venom wakes up from his coma his thirst for revenge triggers Pyscho Mantis to crash a plane that he was being transported in close to Volgin's comatose body. Volgin's hatred for Big Boss controls Mantis and they go to Cyprus to kill Big Boss. The hospital Venom was in was controlled by Cipher in order to hide Big Boss. Part of the plan Zero came up with to hide Big Boss was to brainwash the medic from Ground Zeroes, implant him with Big Boss' memories and perform plastic surgery on him to use him as a body double. Once Venom awoke the hospital staff were instructed to notify Kaz and Ocelot. Somehow XOF (strike force of Cipher now controlled by Skullface) find out about this and mount an attack on the hospital. While Venom was recovering they perform plastic surgery on him to make him look like Big Boss
So you've got this cluster fuck of three forces tearing the hospital apart. Quiet is the first on the scene after Venom's brain washing makes him think that he was seeing Big Boss' face in the mirror instead of his own. She was supposed to assassinate Big Boss, but confused him with Venom. The real Big Boss in the next bed manages to fight her off and gravely wound her. Big Boss and Venom make their way through the hospital but are almost killed by Volgin, who is using Psycho Mantis' power to move and basically make himself an invulnerable walking tank. Big Boss and Venom manage to get away from them. They dodge XOF several times pretty easily (because of Volgin's interference, bulky gas masks, and them trying to seal the hospital and kill everyone in it). Since Big Boss is Big Boss he's able to sneak out and borrow an ambulance to escape in. Volgin (somehow using Mantis' powers to turn a horse into a flaming pegasus) and XOF chase them. Venom and Big Boss crash and are knocked unconscious. Ocelot finds them, drags out Big Boss, gives him Venom's old identification so he can escape Cipher and XOF while building up Outer Heaven. Ocelot then goes back to where Venom is and helps him escape Volgin.
Now a lot of this makes little to no sense and the timeline is kind of weird. We don't know exactly when Big Boss woke up from his coma. It was before Venom, but we don't know exactly when. We also don't know exactly when the brainwashing, memory implantation, and plastic surgery was done to Venom. The game makes it seem like the plastic surgery was done right before the hospital was attacked, but that doesn't make any sense. Venom wouldn't be healed in time. The explanation for why XOF is so slow and blind in the hospital also doesn't really make sense. There were plenty of times where they should have saw you in the game but didn't. This is probably just realism taking a backseat to story and atmosphere. The hospital scene had to serve as a tutorial while also feeling very jarring and weird. And finally, the timeline for Ocelot pulling Big Boss out of the crashed ambulance, giving him Venom's identification, and then coming back also has time problems. He was there with Big Boss near dawn to show him the motorcycle, but somehow when he pulled Venom out of the wreck its still night. And there doesn't seem to be any time where he could sneak away from Venom to do anything. Adding to that, we don't know when Ocelot hypnotized himself. Although it would make sense that he did that on the ship while Venom was in physical therapy and getting back in shape. So take it as you will. For me personally anything in that mission that doesn't fit together great was just in service of atmosphere at the expense of coherence and I'm OK with that. This is the series with cloning, Illuminati type organizations and ghosts after all.
