The number $22/7$ is not irrational, regardless of the number system one uses to express it. On the other hand $\pi$ is irrational, regardless of the number system one uses to express it. They are different numbers: $22/7$ is a good rational approximation to $\pi$, but they are not equal.
It is unfortunate that students are taught to think of "nonrepetition" as the essential quality that distinguishes rational numbers from irrational numbers. It is true that if a number is rational, then its decimal representation will eventually either terminate or repeat, but it might take a very long time, and just looking at a string of digits is not enough evidence to conclude whether or not the string represents the beginning of a repeating decimal or a non-repeating decimal. The real distinction between rational and irrational numbers lies in whether it is possible to express the number as a ratio of integers. If it is possible, then the number is rational; if it is not possible, then the number is irrational. The fact that rational numbers correspond to decimal representations that terminate or repeat is an important and interesting consequence of the definition, but it is not really the essence of the distinction.
For more on this, see my answer and the discussion in the comments beneath it at https://math.stackexchange.com/a/2073186/124095.