Is there a "positive" definition for irrational numbers?

An irrational number is a real number that can be expressed as an infinite simple continued fraction.

A irrational number is a real number $x$ such that for any integer $q'$ there exists a rational number $p/q$ with $q > q'$ and

$$0 < \left|x - \frac pq \right| < \frac{1}{q^2}.$$

If you want simple definition that's not based on Dirichlet's or Hurwitz's theorem, try this:

A real number $x$ is irrational if and only if for all positive integers $n$ there exists an integer $m$ such that $0\lt nx-m\lt1$.

The underlying theorem here is that for all real numbers $x$ and all positive integers $n$ there is a unique integer $m$ (namely, $m=\lfloor nx\rfloor$) such that $0\le nx-m\lt1$. If $0=nx-m$, then $x=m/n$ is rational, and vice versa.