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In Greek and Roman mythology, the Astomi, also known as the Gangines, are an ancient legendary race of people who had no need to eat or drink anything at all. They survived by smelling apples and flowers. Is something like this possible? I mean, we smell things by inhaling particles, and we judge what we should eat by smelling them.
i.e. Can a human-sized species get all their nutrients from the particles they smell?
Your human-sized species might make do with as little as 1000 kCal per day (if they had a fully human metabolism that would be in the neighbourhood of 2000 kCal, but you can posit a slower metabolism).
So we're left with two requirements: nutrients, and energy. Human food supplies both, but here we need some more flexibility.
We can supply energy using suitable vapours.
Gas and alcohol would supply more than enough energy, but that would mean living in a potentially detonating atmosphere. Not recommended. Heavier carbon chains are less reactive, so you could imagine a heavy "fog" of, essentially, heavy Diesel fuel droplets. This could be produced by tree analogues with some equivalent of the Fischer-Tropsch process, and dispersed in the atmosphere, either ubiquitously or in special "feeding ponds". By inhalation, our creatures could easily fulfill their energy requirements; instead of compression, their lungs would simply absorb the fluid and distribute it in the "blood", and the cells would oxidise it much like ours oxidise glucose.
Efficient hygroscopic management and sufficiently humid atmosphere takes care of the drinking needs.
Energy requirements having been taken care of, the quantity of required nutrients will go way down; excepting the growing youngs, pregnant females and those healing from wounds, the adult specimens would need little to no extra mass.
By making growth and healing lengthy processes, we can work around both problems.
Our creatures would inhale somewhat more air than a human would, let's say around 20 cubic meters per day; a possible, if very high, concentration of dust in the air is one full gram per cubic meter (the human COSHH limit is 10 mg/m³ for inhalable dust, but densities up to two orders of magnitude higher have been reported in dust storms and inside containers of specific substances such as flour, maize and sawdust). These densities aren't sustainable for long periods (dust will precipitate and I can't seem to come up with a sensible mechanism to keep it aloft), but the young might romp in the "sawdust", or maybe simply snort it.
Twenty grams of matter per day isn't very much, but that's only for those chemicals that can't be supplied in gaseous form - the "ashes", so to speak. An average adult human being is about 3 kilogram "ashes", and that quantity can be easily inhaled in a couple of years (or as little as five months, if passive inhaling is all that it takes). Our people might be less reliant on heavy and uncommon elements (and be thinner to boot), and only require maybe one to two kilograms of solids.
Having the bones made of carbon compounds (lignin, etc.) would get rid of most of the calcium, which is the large majority in weight of human ashes, and carbon we can absorb from the atmospheric fuel.
Of the main building blocks of organic matter - CHONPS - we can easily supply the first four through respiration and suitable chemical reactions. The relative quantities of Phosphorus and Sulfur are small, and both exist in gaseous form as phosphane, organophosphorous compounds, and hydrogen sulfide.
We only need dust for the remainder.
Of course, at this point the "air" your people breathe is not air anymore, it more closely resembles smoke from a tyre deposit fire mixed with insecticides or nerve gas.
I am afraid it's not possible.
Adults cannot breathe and eat at the same time. So, strictly speaking, whatever we smell ends up in our lungs, not in our stomachs. That would be a first showstopper.
But let's say that these creatures are adapted to this sort of nutrition mechanism. What is the amount of food we need to take into account?
I have found this reference which states
Interestingly, research shows that most humans eat between three to five pounds of food per day. Indeed, as we approach four pounds of food intake for the day, most of us are feeling pretty satisfied.
I have the feeling that 1.2 kg of food per day is quite a lot, but let's take the ballpark figure of 1 kg just as a reference.
When we smell something, we are usually talking about micrograms or even less of substance, dispersed in the whole volume of the environment where we are in.
The lowest odor detection threshold is
Odor threshold value (OTV) (also aroma threshold value (ATV), Flavor threshold) is defined as the most minimal concentration of a substance that can be detected by a human nose. Some substances can be detected when their concentration is only few milligrams per 1000 tonnes, which is less than a drop in an Olympic swimming pool.
Let's say the food of this species is not as rare and to be present in the dilution of 1 gram in 1 cubic meter (1000000 times more concentrate than the quoted example).
To get 1 kg of food, the creatures would need to breathe 1000 cubic meter of air a day, which are 1 million liters. Considering that on average a human exchanges half a liter of air per breath, it would take 500thousand breaths a day, roughly 6 breaths per second, to inhale all that food.
The normal respiration rate for an adult at rest is 12 to 20 breaths per minute.
Wrapping up, no, it's not possible.
Because these creatures are smelling things actively (not just sitting in normal air conditions), they could be sniffing much higher pollen concentrations than one would just hanging out near a tree. During a major pollin bloom, you can see pollen counts in excess of 1000g/m^3; so, your sniffer animals could conceivably "graze" on pollinating plants at that density by sniffing it up at the source.
As L.Dutch points out in his answer, a human sized mammal can live off of 1.2kg of food per day which means the animal only needs to breath in 1.2m^3 of air under these conditions to sustain itself. That is 2400 half liter breaths which would take a normal human about 13.3 hrs per day to breath in at a normal respiration rate, but an animal adapted for inhaling lots of air for purposes other than just breathing could probably do it much faster.
While this is mathematically feasible, that means your animal would need to have a constant supply of blooming pollin year around... if it is a mammal. Luckly, there are animals that can live off of way fewer calories than mammals: reptiles. Reptiles need ~7.2 times fewer calories than mammals. Reptiles can also survive for months on end without any food at all. The way I would solve this problem is that your creatures are reptiles that actively go around sniffing flowers for about 2 months during spring when everything is blooming kicking off enough to reach these really high pollen counts. During these months, your reptiles can breathe in enough calories to sustain themselves for the year, then the other 10 months, they sit around like sticks in the mud conserving calories waiting for the next bloom.
Reptile physiology works out especially well here because they can be just as active in spring as mammals without nearly the calorie cost because they are not spending the energy on regulating body heat. This means they can move from flower to flower as needed.
A hibernating mammal might be able to pull this off as well, but mammals generally can not hibernate for more than half the year; so, it would either need to be very specialized or live in an environment with plants that have spread out heavy pollen blooms.
A tablespoon of bee pollen have around 40 calories. 50 000 of those and you're set for a day (assuming requriment of 2000 kcal a day).
Adult human, while resting inhale around 11000 litres of air per day. So you would need 4,5 teaspoon of bee pollen in one litre of air. With each breathe we inhale around 0,5 litres of air. Imagine breathing two tablespoons of fine powder. Remember cinnamon challenge?
The problem was the amount of particles that could clog the lungs.
I think this could exist in those middle ages books about "far-away" land where people were made out noses entirely. So the surface to filter and somehow consume particles was much bigger and easier.
Think about it more creatively. Consider if this being was a plant instead of an animal. On Earth, we think of plants as generally being non-mobile, but I don't think that is a requirement. Remember that some orchids live without drawing nutrients from their roots. "Nutrients for epiphytic orchids mainly come from mineral dust, organic detritus, animal droppings and other substances collecting among on their supporting surfaces."
