I would say there are two challenges. The first one is how to represent the data under consideration?. The second one is the choice of the classifier according to the nature of the data?.
I would say that 100% accuracy bis not viable unless the underlying distribution is known. So this makes complete accuracy unachievable. That being said there is still much we can do in terms of representation as Sabeur said, and extracting relevant information from the patterns given.
It is possible to achieve 100% accuracy if the classes are different enough: for example, you could build a classifier that separates astronomical objects into stars and asteroids based on the mass of the object. On the other hand, sometimes it impossible to build a classifier with 100% accuracy as there is overlap between the classes (you cannot classify between men and women based on height no matter what kind of classifier you use.) Then there is the question of measurement error present in real data.
To summarize, it is possible to build a classifier with 100% accuracy if the distance between classes is much larger than the within class variability + measurement error.
Yes, it is possible to achieve 100% accuracy if the classes are different enough but the assumption has to be made that the underlying distribution behaves like that and that is a strong assumption for classifiers in general.
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