All you have to do is row reduce! Put everything into one matrix, and get rid of as many rows as you can.
Logically, in a basis, first of all there cannot be the zero vector. Secondly, no vector in a basis can be a linear combination of any other vectors. In your example, $X_4 = X_3 + X_2$, so you can leave out $X_4$. You are left with your basis: $X_2, X_3$
If you want to row reduce with a more complex question, you have the following matrix: $\left[ \begin{array}{cccc} 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 1 \\ 1 & 1 & 0 & 0 \\ 1 & 1 & 1 & 1 \end{array} \right]$
Move the zero row to the bottom for convenience.
$\left[ \begin{array}{cccc} 0 & 0 & 1 & 1 \\ 1 & 1 & 0 & 0 \\ 1 & 1 & 1 & 1 \\ 0 & 0 & 0 & 0 \end{array} \right]$
Now subtract the first row + the second row from the third row.
$\left[ \begin{array}{cccc} 0 & 0 & 1 & 1 \\ 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array} \right]$
We cannot row reduce anymore. Your answer is the rows that are not completely 0's. The first row, which maps to $X_2$, and the second row, which maps to $X_3$, is your basis. Remember, for the basis, you should use the original vectors, so the original $X_2$ and $X_3$ (coincidentally they are the same this time).
So your basis is $X_2, X_3 = \begin{bmatrix} 0 \\ 0 \\ 1 \\ 1\end{bmatrix},\begin{bmatrix} 1 \\ 1 \\ 0 \\ 0 \end{bmatrix}$