Nietzsche is fond of hyperbole and metaphor, and it might not be immediately apparent what he means when he accuses the majority of his contemporary Europeans of being "sick." In the last decade of his working life, when the Genealogy was written, Nietzsche himself was very sick, suffering from migraines, insomnia, and near blindness among other things. Nonetheless, he felt himself to be in a far greater state of health than most of his contemporaries who, though healthy in body, were sick in mind and spirit.
Nietzsche claims that this "sickness" arises from the constant struggles and torments that we put ourselves through. We have gained depth, morality, society, an inner life--everything that we might claim distinguishes us from animals--through self-torture and struggle. We could go so far as to say that we are the "inward-looking animal," and that this inward looking has only been generated by a constant struggle against ourselves and our own nature. The greatest triumph, for Nietzsche, is to delight in and affirm this self-torture and struggle, to see it as a willful act of creation, whereby we free ourselves of our instincts and our evolutionary past, and fully create ourselves. More often than not, however, we do not see all our torments as a triumph, but rather look upon them as sufferings to be endured. If we see life as suffering, life becomes something to be pitied, something that might arouse nausea. This pity and nausea are what Nietzsche denotes as the great "sickness" in humanity. Those who become sick of humanity are not strong enough for the struggle that is humanity. From this sickness grows ressentiment, nihilism, and everything else Nietzsche despises.
"Sickness" is an apt name because it is contagious. It generates a slave morality that persuades the strong that they are evil, and induces them to self- hatred and sickness as well. The only safety for the strong is in avoiding the sick masses and ignoring their moralizing.
The ascetic ideal among the masses is an expression of a sick will to power. The sick are suffering from life, seeing life as a misfortune, and in the ascetic ideal they find a means of asserting themselves. Any positive act of will (pursuing health, happiness, strength, etc.) is beyond their means, and so they cannot will these things. Instead, they will nothingness, the only thing they can successfully will. As Nietzsche claims at the beginning of this essay, the sick would rather will nothingness than not will.
There is no doubt that Nietzsche opposes the "sick" as a bad thing and antithetical to life. However, we should not fully associate asceticism with sickness. Nietzsche finds only one interpretation of asceticism in claiming it is the only expression of the will to power left available to the sick. Asceticism is only bad insofar as it might be indicative of sickness. However, this is not the only one way of looking at asceticism: we have already seen that Nietzsche gives it different meanings for philosophers and artists.
That being said, we should also note that Nietzsche considers the artist's asceticism to be found in the philosopher's asceticism, and the philosopher's asceticism to be related to the ascetic priest's. In that sense, they are all in some way indicative of sickness, but the matter is more complex than a simple "asceticism is bad."