Is there any single element (e.g. possibly carbon) that can be eaten, which has a nutritional value such that it provides calories?
I found on Wikipedia (dietary element) that there are certain elements that our bodies require. I suppose oxygen could count as such a nutritional element, although we breathe it instead of eat it.
Interesting question! You may get a better answer, but I believe the answer is no.
Calories are units of heat and represent energy available to the body. The form of fuels that can be (biochemically) "burned" by the body are hydrocarbons; i.e. molecules of carbon and water, like sugars. Carbon alone I do not believe can work.
The only other single element fuel is hydrogen, and again I do not think this is biologically available to the body. By the way, oxygen is required as the oxidizer in any combustion reaction, and we definitely need oxygen (from breathing) to convert the food we eat to energy. But oxygen is not a fuel: it is the oxidizer that is reduced during combustion.
While your question refers to human metabolism, it should be noted that bacteria can use elemental hydrogen, elemental sulfur, and elemental iron as energy sources:
See Wikipedia's page on Microbial metabolism, particularly the section on chemolithotrophy.
If any of these bacteria are found in the human gut, and their metabolic processes support our own (as is the case for symbiotic gut bacteria), then effectively we can use these elements for energy as well.
Caloric value is basically stored energy. Plant-based foods create this stored energy through the absorption of sunlight, and in some extreme situations, I imagine there may be plants that generate energy from thermal vents and such. They then store that energy in a molecular combination that you can think of as compressing a spring, giving it potential energy.
The same goes for animals, whereas rather than sunlight that energy is created from eating either plant or other animal life.
Mass inherently has energy only at the mass-energy equivalent level, and biologically we are not capable of accessing that.
From the conservation of energy, if the element you are putting in your body is not part of a molecular "energy spring" as per the analogy above, it will not spontaneously posses energy just because you are ingesting it.
As far as Carbon, this is one of the great misunderstandings about fuel. We burn Hydrocarbons as a fuel source. Wood originally, then coal, and now we have jet fuel. They are basically all the same thing, with decreasing Carbon to hydrogen ratios. The energy stored is actually being released from the hydrogen bonds. The ideal goal is for us to be using carbon-less hydrocarbons as fuel sources as the only burned waste is water as it combusts with the air, no CO2 providing the highest weight to fuel ratio (no carbon along for the ride reduces the weight) and doesn't pollute.
Most metals consumed (albeit in only tiny amounts for select elements) will be directly converted to ions by the HCl in our stomach.
Then, as the question includes the word 'nutritional', commonly cited elements (in their ionic form) of particular import include: potassium/sodium (regulates fluid balance, muscle contractions, and nerve signals), iron (linked to hemoglobin and the transportation of oxygen in the blood), calcium/magnesium (maintains strong bones), iodine (required for creation of thyroid hormones and regulating the body's metabolism), chromium (important in the breakdown of fats and carbohydrates) and copper (instrumental in formation of red blood cells, maintenance of nerve cells and the immune system), to mention a few.
Of the mentioned elements, those having a caloric impact include chromium and iodine.
Note, elemental carbon, especially in the form of charcoal pills or activated carbon, should be avoided (not nutritional or even caloric), see discussion here.
Further, I would argue, that as the human body generates a chemical-based electric current (see this discussion), it is probably not wise to introduce a very good cathode (namely carbon). Unwanted solvated electrons and radicals created therefrom, most likely do not promote good health, and in ones' body, may supply an electrode to promote electrochemical (or battery) cell activity.
