I’m referring to an entomological plastron which is a structure that traps a bubble of air next to the insect's body but permits contact between the air and the water. As the insect consumes oxygen and releases carbon dioxide the carbon dioxide diffuses out of the bubble into the water and oxygen diffuses in. The rigid structure of the plastron makes the air bubble incompressible and therefore permanent. In this way, the plastron acts as a sort of artificial gill. As long as the surface area of the bubble is of sufficient size to allow diffusion to keep up with the needs of the animal it will never have to surface.
Now, my question is how large (in surface area) does a plastron or artificial gill need to be to accommodate the oxygen needs of a human indefinitely? Since I imagine it will be large let’s assume it isn’t mobile but rather acts as a stationary structure that the human can swim in and out of (with an airlock of course). Essentially, imagine an undersea base where the walls are rigid and gas permeable.
One other minor issue with this system is that all of the nitrogen in the plastron will gradually be replaced with oxygen and this, combined with the increased pressure of being underwater, will lead to a highly toxic oxygenated atmosphere. For my purposes I don't care because I’m actually designing an alien species of spider-like creatures who weave rigid gas-permeable cocoons deep underwater inspired by the Diving Bell Spider. My species will have evolved to tolerate these high oxygen environments but will have similar metabolic needs to humans and I feel it makes the question much easier to answer and potentially useful to others if we leave it as humans.