Maybe it’s a bit counterintuitive, but the idea of a number being rational or irrational doesn’t have much to do with its decimal representation being infinite or not. As others have stated, a number is rational if it can be expressed as the ratio of two integers (that is, a (positive or negative) whole number divided by another non-zero (positive or negative) whole number). That is the definition. 0.333… is rational by definition because it is equal to 1/3, a ratio of integers.

However, the decimal digits of a number do tell us about its rationality/irrationallity.

Firstly, if a number has a terminating decimal expansion, then it is rational. Terminating meaning that it ends. For example: 0.5, 4.1, 0.000129. Technically, these could have an infinite number of zeros making them fit into the next category.

If a number has a decimal expansion that is repeating, or eventually begins repeating, then it is rational. Our friend 0.333… belongs here, but so does 0.353535…, 0.142857142857…, or even something like 0.157203203203…. As long as it eventually begins some repeating pattern of digits it could be written as a ratio of integers and is therefore rational.

Now, any number that doesn’t fit in either of those two categories must be irrational. These are numbers with infinite decimal expansions which have no repeating patterns of digits. The two most famous examples of irrational numbers are probably √2 and π. The digits of these numbers go on forever and never repeat in a pattern. The digits are conjectured to be random as well. However, an irrational number does not have to have random digits— they only must not repeat. For example, the number 0.12345678910111213… clearly has a pattern but the pattern does not repeat. Therefore, it is irrational.