The diagonal is long by 1 m + 4 dm + 1 cm + 4 mm + 213 µm + 562 nm + 373 pm + 95 fm...
You indeed accumulate "an infinity" of lengths, but the sum converges very quickly and is finite (not counting the fact that you quickly get to the size of the quarks).
The "non-repeatingness" is justified by the fact that any repeating number can be expressed as a fraction.
For instance, 1.4142141421414214142141421... is 141420/99999. (Take that number, multiply it by 100000 and subtract the original; all decimals cancel out.)
And $\sqrt2$ cannot be a fraction. If it were, let $\sqrt2=\frac pq$, with $p$ or $q$ odd (if both are even, you can simplify). Then $p^2=2q^2$, so that $p$ is even (the square of an odd number cannot be even). $p$ being even, $p^2$ is a multiple of $4$, so that $q^2$ is even, and $q$ is even !
As $\sqrt2$ cannot be a fraction, it cannot be a periodic number.