The logic table: $$\begin{array}{|c3:c|}\hline x & y & z & f(x,y,z) \\\hline 1 & 1 & 1 & 0 \\ 1 & 1 & 0 & 1 \\ 1 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 \\ \hline\end{array}$$
The DNF I came up with based on this table is $f(X,Y,Z) = (X \land Y \land Z) \lor (\lnot X \land \lnot Y \land Z)$
Here is my attempted solution at a combinatorial circuit:
Is this the correct way of doing it for this logic table? If not, where did I go wrong?

The "Logic Gate" diagram is okay for what you have except that is not how you draw an "or-gate".
However, your function is not okay. Notice that you have $f(1,1,0) = (1\land 1\land 0)\lor(0\land 0 \land 0)$ which is $0$. You need it to be $1$.
$$f(X,Y,Z) ~\ne~ (X \land Y \land Z) \lor (\lnot X \land \lnot Y \land Z)$$
It is close though. (Maybe a typo?)