With the growing awareness to fairness in machine learning and the realization of the central role that data representation has in data processing tasks, there is an obvious interest in notions of fair data representations. The goal of such representations is that a model trained on data under the representation (e.g., a classifier) will be guaranteed to respect some fairness constraints. Such representations are useful when they can be fixed for training models on various different tasks and also when they serve as data filtering between the raw data (known to the representation designer) and potentially malicious agents that use the data under the representation to learn predictive models and make decisions. A long list of recent research papers strive to provide tools for achieving these goals. However, we prove that this is basically a futile effort. Roughly stated, we prove that no representation can guarantee the fairness of classifiers for different tasks trained using it; even the basic goal of achieving label-independent Demographic Parity fairness fails once the marginal data distribution shifts. More refined notions of fairness, like Odds Equality, cannot be guaranteed by a representation that does not take into account the task specific labeling rule with respect to which such fairness will be evaluated (even if the marginal data distribution is known a priory). Furthermore, except for trivial cases, no representation can guarantee Odds Equality fairness for any two different tasks, while allowing accurate label predictions for both. While some of our conclusions are intuitive, we formulate (and prove) crisp statements of such impossibilities, often contrasting impressions conveyed by many recent works on fair representations.