What happens when someone is stung by a jellyfish?
The problem is that box jellyfish doesn't specify one jellyfish but a group of different jellyfish. Some of these are highly venomous - I pick here Chironex fleckeri, as this is often called "the most venomous jellyfish in the world".

Chironex fleckeri has long tentacles which are covered with millions of explosive cells called Cnidocytes which inject a dart with the very powerful toxin upon touch. It is therefore very dangerous to touch the tentacles with bare hands to remove them.

The toxin is a mix of different bioactive proteins which have cytolytic, cytotoxic, inflammatory or hemolytic activity. Case reports (you can find information in reference 3 about it) show that the tentacles "burn" though all skin layers causing immediate strong pain and lasting scars (if the victims survive) which tend to show signs of necrosis. The toxins itself paralyze the muscles of the heart and the respiration and also causes hypokalemia by hemolyzing red blood cells (one of the toxin acts as a membrane pore in the blood cells). This causes further problems and often a cardiac arrest. See reference 1 and 2 for details on the toxins. Especially 2 is interesting, as this is a PhD-Thesis on this topic with a lot of references.

Treatment depends on how severe the injury is - in severe cases an antivenom can be used. Besides this typical measures are CPR, giving oxygen and removing the tentacles (with appropriate protection). See reference 3 (good overview) to 5 for details on this topic.

References:


Chironex fleckeri (Box Jellyfish) Venom Proteins: Expansion of a
Cnidarian Toxin Family that Elicits Variable Cytolytic and
Cardiovascular Effects
The molecular and biochemical characterisation of venom proteins
from the box jellyfish, Chironex fleckeri
Chironex fleckeri (Multi-tentacled Box Jellyfish)
Jellyfish stings and their management: a review.
Cytotoxic and cytolytic cnidarian venoms. A review on health
implications and possible therapeutic applications.