To summarize what ketosis is, as shown in the nice summary graph you included, there is 3 main types of nutrients we can use: carbohydrates, fats and proteins.
Carbohydrates can be broken down and converted into glucose (well, not all, such as many of the components dietary fiber, since we lack the enzymes to break these down. This is why if we eat wood, which contains a lot of cellulose, we do not actually get any nutrients from it). For long-term storage, we can use glucose to generate a polymer known as glycogen, which is stored in the liver (but not in other organs, such as in the brain).
Fats are broken down mainly into fatty acids, which can be broken down further into a molecule called acetyl-CoA, which can be used to obtain energy (in fact, when obtaining energy from glucose, we also need to go through the intermediate step of conversion to acetyl-CoA). For long-term storage, fatty acids are merged with glycerol to form triglycerides, which are the main form of storage in fat cells. This is in fact how we store most of the excess energy coming from the nutrients we consume, and it is the main constituent of "body fat".
For this discussion, I think we can leave proteins as a source of energy out (at least for the time being), since they cannot be used as the main source of energy over a long period of time (we would need to consume very large amounts of protein to cover energy needs just from protein, and that would lead to kidney and liver damage). The main purpose of eating proteins is to break down into amino-acids, which we then use to produce our own proteins (basically, almost every functional thing inside our cells is a protein).
Our brain cannot use fatty acids as a source of energy, because fatty acids are delivered to the organs for use as an energy source bound to a large molecule (a protein) called albumin. The reason is that they need to be delivered through blood, but fatty acids are not soluble in aqueous solutions (which blood, and the intracellular medium, are). Albumin, however, is soluble in aqueous solutions, so by binding fatty acids to it, we can force them to be soluble. If not, the fatty acids would just clump together, like when we try to dissolve oil in water.
Why does this matter? Well, our brain is a special organ (!), and therefore it has much more protection layers than other organs. For example, the so-called blood-brain barrier. This is basically a physical layer formed by a specific type of cells that tightly controls (much more tightly than what happens for other organs) what goes from the blood stream into the brain. This is essential for our survival: for example, it helps to keep pathogenic microorganisms away from the brain. But it also creates problems, since it also means many molecules (specially large molecules) that can reach other organs, cannot reach the brain (in fact, it is one of the main consideration that needs to be taken into account when designing medicines that are targeted to the brain). Amongst these is albumin, but not glucose.
So what happens under normal circumstances, is that our brain is fueled by the glucose in food we eat. If we have not eaten in a while, our liver breaks down the stored glycogen (the polymer of glucose), the resulting glucose passes to the bloodstream, and then goes to the brain. By doing so, we don't need to be constantly eating things that have glucose to keep our brain working.
However, the amounts of glycogen we can store are limited, and they will run out in a few hours (shorter time if you do intense exercise). We usually don't black out even if not eating in a few days, so what's going on here? You might think maybe we are making glucose from the large stores of triglycerides that we have in our body fat. But humans cannot make glucose from triglycerides, fatty acids, or any derived molecule (plants, for example, can do so). Instead, what we can do is to produce the so-called ketone bodies from acetyl-CoA (which can be readily derived from fatty acids). Ketone bodies are in fact a range of several molecules, the main ones being acetone (yes, the same as the organic solvent!), acetoacetate and 3-hydroxybutyrate. These molecules are not the most efficient way to store energy, but they have an advantage, which is that they can readily go into the blood stream and cross the blood-brain barrier. So they can be used by the brain as a source of energy, and this is how our brain survives in the long-term if we do not consume any food.
This process of producing these ketone bodies (I do not really like the name by the way, because it makes it sound like some sort of big-sized inclusions, while they are just small molecules, but the name is widely accepted and used) is what is called ketosis. You might have read that ketosis is good for weight-loss, and the reason is that if our body starts to produce at large scale these ketone bodies, it is doing so at the expense of body fat.
However, it comes with consequences. Nutrition websites frequently called this "the keto flu", because some of the symptoms are similar to those of flu. These include headaches, increased irritability or change of mood, digestive tract disorders, dizziness, amongst other. It is true these can go away to some extent, but the extent of the symptoms, how much and when they disappear varies. There is in fact quite a lot of research about the long-term effects of ketosis still going on, with some suggesting even an increased risk of cardiovascular disease.
I have not been able to found (on a quick search) a paper directly analyzing if ketosis can make normal cells be more resistant to ionizing radiation, but there seems to be quite a bit of research on the combined effects of ketosis and radiotherapy on cancer cells. See for example this, this and this recent review. These papers seem to suggest that the sensitivity of cancer cells to radiation therapy is in fact increased by ketosis. The reasons seem to be unclear, and might be partially related to an effect of ketosis on the production/elimination of the so-called reactive oxygen species (ROS). ROS are unstable molecules derived from molecular oxygen that are a full different subject on its own, but in short they are involved in aging, inflammatory processes, mutations in the genetic material that lead to cancer, and many other "bad" processes. So does ketogenesis promote or reduce the generation of such ROS? From my understanding, it is unclear. For example, quoting from the mentioned review, paper:
Finally, the antitumor effect of the KD has been associated with both increased and reduced ROS levels, indicating that the KD potentially interferes with the tumorigenic ROS balance of cancer cells
Where KD means ketogenic diet. So basically it is not known! Maybe the increased sensitivity of cancer cells to radiotherapy is just a combination of the radiotherapy itself and the negative effects of ketosis on cancer cells, and not necessarily a sign that it makes cancer cells more sensitive to radiation... In any case, I have not found any strong evidence to suggest that ketosis makes cells more sensitive or more resistant to radiation.
A diet that induces ketosis (i.e., one where foods with carbohydrates are eliminated to a very large extent) also needs to be very carefully planned, specially if aimed to be sustained in the long term, since we also need vitamins and other micronutrients (nutrients that we need in very small amounts, not for energy production, but in other for components of our cells to be functional). By strictly eliminating foods with carbohydrates (including fruits) from the diet, we might be also removing sources of said micronutrients. Of course, I'm sure if a ketogenic diet was going to be implemented by any space agency, this would certainly be taken into consideration, but I thought it was worth mentioning that this also needs to be kept in mind, since it might mean that a very strict ketogenic diet is difficult to sustain in the long term.
So given the lack of evidence to support an increased resistance to radiation, the presence of quite negative side-effects and the fact that the diet might lead to micronutrient deficiencies if kept very strictly ketogenic, I do not think it sounds like a very likely choice for people potentially living on Mars. Especially given that this would be over a relatively long period of time. And particularly, with consideration to the ability to withstand small spaces, given that ketosis can lead to irritability and mood changes, it might even be the opposite. Another side effect is that when ketosis happens, we exhale the ketone bodies molecules through our breath (including acetone). On Earth, this does not matter much, since it just dissipates onto the atmosphere. But in closed spaces, having increasingly large concentrations of acetone and acid molecules in the air, might be something to worry. It would certainly require consideration. As a side note, I recently saw a product being advertised as a hand-held, portable device able to enable you to hack your metabolism by simply analyzing your breath; I'm quite sure this is to a large extent simply analyzing the concentration of acetone or other related molecules on the air we exhale.
It doesn't seem to be a good idea...
I have also found this (quite long) report by NASA about the particularities of nutrition in space exploration. Ketosis is briefly mentioned in pages 12 and 14, and it does not look like they thought of it as a great idea. Quoting from page 12:
The metabolic condition of ketosis, which would be expected to result from starvation, not only would have metabolic effects (including decreased appetite), but might also affect other aspects of the mission (for example, the life-support systems might not
be able to remove the ketones from the air). Ketoacidosis can obviously have negative effects on acid-base balance, which in turn can affect bone, muscle, and other systems.
And from page 14:
A ketotic state would likely impair performance of crewmembers, as seen in studies conducted by the military (71), as
well as increase renal stone risk secondary to reduced urinary pH (87-89). Other aspects of the mission would also be at risk (for example, the life-support systems may not be able to remove exhaled ketones from the air). Toxicity of carbohydrate has not been
well studied, and would likely be an issue only because it would displace other nutrients (protein and fat) from the diet.