Constructing the natural numbers without set theory. As I understand it the idea of defining everything as sets is a relatively new idea in mathematics.  Does that mean there's a non-set theoretic definition of the natural numbers?  Could there be?
One way to construct the natural numbers (without sets) is type theory. We define an inductive type $\mathbb{N}$ with two constructors:

$$
1 : \mathbb{N}
$$
$$
S : \mathbb{N} \to \mathbb{N}
$$

Where $S$ "adds one" to a number. In this system $2$ is represented by $S(1)$, $3$ is represented by $S(S(1))$ etc. This is essentially a more direct implementation of the Peano axioms.
Yes, there is Peanos construction of them which is acctually my favourite method for constructing them.