The outer layer of skin of a human is called the epidermis. It is supposed to be waterproof, however:

Using any current science and substances, would it be possible for humans to survive for two months purely by absorbing substances through the epidermis? The skin must not be pierced or otherwise physically damaged. (Prune skin does not count as damage)

There is no limit to how many hours per day it takes. The subjects lie completely submerged in a bath of nutrients with breathing apparatus. They can get in and out but must not use any other source of food or water.

Clarification

The subjects are ordinary humans who are concerned about ageing. They have volunteered for an experiment. The crazy scientist has convinced them that if they stay there for two months (sedated to avoid claustrophobia but with VR glasses for entertainment) they will regain their youth. Assuming they can stand the discomfort, will they make it out alive or will they die or have to be rescued?

Assuming they can stand the discomfort, will they make it out alive or will they die or have to be rescued?

They will starve.

Why?

Nutrients and water are absorbed in the intestine.

The average surface area of the human intestine is about $32 m^2$.

The average surface area of human skin is $1.5$ to $2.0 m^2$.

The average transit time in human intestine, from ingestion to expulsion, is about 50 hours.

This means that your subject will die of malnutrition, because they have 16 time less exchange surface to get their needed nutrients. To get the same amount of nutrients of a normal digestion they would need to bathe $16 cdot 50 = 800 hours = 33 days$.

Death by starvation takes about that time, death by dehydration much less.

NO

The skin is a very specialized organ. Its job is litterally to be the first and greatest line of defense against the environement. It is what allows other organs to do their jobs without having to deal with for example dirt suddenly being on them. Afterall division of labor is what allows for specialization. So naturally it makes sense that skin is designed to block out as many substances as possible, like putrid water from a lake. Meanwhile other specialized structures are designed to allow the inteded entry of substances like a mouth.

Water

Between the two options water is more plausibly absorbed by the skin. Problem is it cannot sustain life. Skin is designed to block substances as snall as the nano scale. Employing everything from fats and proteins to form such barriers. Water is a very small molecule so yes it can permeate these though at a slow rate.

The problem is the body expells excess heat and waste at great expense to its water supply. Thus it uses it faster than it can absorb it through the skin.

Food

Utterly impossible. Some nutrients can permeate the skin. But food, the bulk majority of nutrients cannot. They are simply too large. More over most nutrients need to be converted in the intestine to be usable to the body, otherwise they would be out rigjt toxic.

In this situation we have two main thing to address: water and food.

• 
  

  Food.

  
  
  
  
  
  • They starve. Taken persons health, body mass, body fat, people can starve for 3-12 week, with some cases for 4-6 months.
  
  
  • As there is no info on this way of feeding, it is just a leap of thought. You could add keton bodies, maybe with ethanol ass addition.That will help them fuel ketosis.
  
  
  
  

  And yes, you can get drunk from bath in booze and die. Ketones do get absobed by skin. Just be carefull and not kill your patient, as too much is death.

  
  
  

  You can try adding glucose and BCAA. Note all that, most likely, wont be close to needed so you will be in starvation mode, but it will give you time and more comfort.

  
  


• 
  

  Water

  
  
  
  
  
  • Is water absorbtion from bath, assisted by ethanol, enough? More data needed. You can adjust air-supply so they loose less water, with sweat gone you can save close to 1 liter of water. Will you get 0.7-1.5 liters needed? Maybe.
  
  
  
  

In the end, if our mad scientist finds out, that his measures did not work. Just let them drink water and starve. Healthy with some body mass to lose will make it, weak with no spare body mass will die.

I will not volunteer for this one.