Question: So I know that set $S=\{(x,y,z)| x,y,z\in \mathbb{Z}\}$ is a subset of vector space $\mathbb{R}^3$.

Specifically, it is worded in our lecture that it is a " subset of $(\mathbb{R}^3, \oplus, \odot)$ , where $\oplus$ and $\odot$ are the usual vector addition and scalar multiplication."

My teacher has stated in our lecture that this set $S$ is not a subspace of $\mathbb{R}^3$.

But from what I can tell $S$ is:

1. Closed under addition
2. Closed under multiplication
3. Contains a zero vector $(0,0,0)$

How is it not a subspace of $\mathbb{R}^3$, what am I missing?

Answer: 
$(1,0,0)$ belongs to $S$. Scale this by the real scalar $1/2$ and obtain $(1/2,0,0)$ which is not in $S$. Hence $S$ is not a subspace.
It is not closed under multiplication. 
Take $\lambda=\sqrt{2}$ and any vector $v=(a,b,c)\in S$. Since $a,b,c\in \Bbb Z$ then $a\sqrt{2}, b\sqrt{2}, c\sqrt{2}\notin \Bbb Z$. Then $\lambda\cdot v \notin S$.