When talking about self-sustainability re: food, a healthy diet is paramount, and, in my opinion, not difficult to achieve – unless one constrains oneself, that is.
Right out of the gate, veganism is not a smart choice in this regard IMO (and it is only this regard that I am talking about now). Despite all the good arguments for veganism, it is often a luxury that not everyone can afford, and it is especially expensive when one pays for the nutrition with back-breaking labor. And fifth-level vegans would starve, since even potatoes cast a shadow ( I never liked that joke; it clashed with my literal-mindedness).
The biggest problem is, of course, vitamin B12. There are basically only two ways of getting it outside of eating animal products. One is to take supplements, which is not possible in a self-sustainable way. The other way is to use feces to fertilize vegetables, and not being too thorough when washing them for cooking afterward. Eating your own poop works for lagomorphs, but I would not be particularly thrilled about doing it.
The second problem is essential amino acids. There are plants that contain complete proteins, like potatoes and soy, and there are combinations of plants that, whilst having incomplete proteins each, complement each other, like beans and corn. However, all of these do come with some drawbacks – potatoes contain very little protein, all legumes contain chemicals that inhibit protein digestion, and any plant combination requires additional knowledge and work. There are workarounds for these drawbacks, but they sometimes do not scale down well from an industrial process to small-scale self-sustainability attempts (like fermenting, making tofu etc.).
The third problem is iron and calcium. Both of these are needed in relatively large amounts compared to other micronutrients, and both are contained in large-ish amounts in some plants, so intuitively, there should not be a problem here. But there is, because the large fiber content of fruits and vegetables makes both iron and calcium less bioavailable compared to animal products. Which also dovetails into the last problem.
Plant-based foods require more energy to digest than many animal-based products for the same caloric/nutritional gain. When doing a lot of hard labour, a vegan would need to eat (and grow) a bit more food just to keep up.
On the other hand, a strict carnivorous diet is an even less wise choice in this particular context. Although animal-based foods do contain the full gamut of nutrients and thus require less knowledge and processing, they do come with many health risks. All the way from constipation due to lack of fiber, across gout to heart disease. Animal-based foods also tend to be more difficult to preserve, and when they spoil, they are very dangerous to health. And these significant health negatives notwithstanding, even if one does not speak about an area that is sufficient enough for hunting and fishing throughout all seasons, one would still need to grow and store plant material for the animals to survive winter. So much so that it would be actually orders of magnitude more labor and area-intensive than directly growing said plant material suitable for direct human consumption.
So which diet lends itself best to an attempt at self-sufficiency? Omnivorous diet heavily weighted towards the vegetarian end, with fish, eggs, dairy, or occasionally poultry, and rabbit thrown in. Such a diet would fall at an optimum with regard to both labor and land use – orders of magnitude less land and labor than a carnivorous diet, and slightly less land and labor than a vegan diet. The animals could utilize the waste and offal that are not suitable for human consumption, and/or they could utilize marginal lands, where food for direct human consumption does not perform well.
And it is concerning such a diet that I will write my subsequent thoughts on the matter.

Many people who attempt to be strict carnivores forget to eat the *whole* animal, including eyeballs and stomach contents.
I’d rather eat my vegetables.