Very broadly speaking, there are two reasons someone would build an automatic classification system (a machine learner, if you will). Either you want to do better than an incumbent manual approach (such as a human expert or group of experts) or you want to do at least as well as the current approach, but use the automatic system to make the classification process faster and be able to handle large amounts of data.
The different applications then drive the acceptable error rate. If you are trying to improve the state of the art, then you need to demonstrate consistently better accuracy than what is currently achieved. Anything better than a benchmark is 'acceptable'. Alternatively, if you are hoping to automate some process currently done by a human then you might only need to do roughly as well as the benchmark since the benefits flow from the automation rather than accuracy boost.
I guess a third option is the case where there is no current system in place at all. In that case as suggested by Michael Chernick, an acceptable error rate could be derived by assessing the cost of misclassification compared to the utility gain from the successes. Again, the application the system is being built for is the driver, there are no absolutes.