Select points $T_1,T_2$ and $T_3$ in $\mathbb{R}^3$ such that $T_1=(0,1,0)$, $T_2$ is at the origin, and $T_3=(1,0,0)$. Let $T_0$ be a point on the line $x=y=0$ with $T_0\neq T_2$. Suppose there exists a point $X$ in the plane of $\triangle T_1T_2T_3$ such that the quantity $(XT_i)[T_{i+1}T_{i+2}T_{i+3}]$ is constant for all $i=0$ to $i=3$, where $[\mathcal{P}]$ denotes area of the polygon $\mathcal{P}$ and indices are taken modulo 4.  What is the magnitude of the $z$-coordinate of $T_0$?
1. **Define the coordinates of the points:**
   - \( T_1 = (0, 1, 0) \)
   - \( T_2 = (0, 0, 0) \) (origin)
   - \( T_3 = (1, 0, 0) \)
   - \( T_0 = (0, 0, z) \) (since \( T_0 \) is on the line \( x = y = 0 \) and \( T_0 \neq T_2 \))
   - Let \( X = (x, y, 0) \) be a point in the plane of \(\triangle T_1T_2T_3\).

2. **Calculate the areas of the triangles:**
   - The area of \(\triangle T_1T_2T_3\) is given by:
     \[
     [T_1T_2T_3] = \frac{1}{2} \left| \begin{vmatrix}
     0 & 1 & 0 \\
     0 & 0 & 0 \\
     1 & 0 & 0
     \end{vmatrix} \right| = \frac{1}{2} \left| 0 \right| = \frac{1}{2}
     \]

3. **Express the condition given in the problem:**
   - The quantity \((XT_i)[T_{i+1}T_{i+2}T_{i+3}]\) is constant for all \(i = 0\) to \(i = 3\).
   - This implies that the product of the distance from \(X\) to \(T_i\) and the area of the triangle formed by the other three points is constant.

4. **Calculate the distances and areas:**
   - For \(i = 0\):
     \[
     (XT_0)[T_1T_2T_3] = \sqrt{x^2 + y^2 + z^2} \cdot \frac{1}{2}
     \]
   - For \(i = 1\):
     \[
     (XT_1)[T_0T_2T_3] = \sqrt{x^2 + (y-1)^2} \cdot \frac{1}{2} \left| \begin{vmatrix}
     0 & 0 & z \\
     0 & 0 & 0 \\
     1 & 0 & 0
     \end{vmatrix} \right| = \sqrt{x^2 + (y-1)^2} \cdot \frac{1}{2} \cdot z
     \]
   - For \(i = 2\):
     \[
     (XT_2)[T_0T_1T_3] = \sqrt{x^2 + y^2} \cdot \frac{1}{2} \left| \begin{vmatrix}
     0 & 0 & z \\
     0 & 1 & 0 \\
     1 & 0 & 0
     \end{vmatrix} \right| = \sqrt{x^2 + y^2} \cdot \frac{1}{2} \cdot z
     \]
   - For \(i = 3\):
     \[
     (XT_3)[T_0T_1T_2] = \sqrt{(x-1)^2 + y^2} \cdot \frac{1}{2} \left| \begin{vmatrix}
     0 & 0 & z \\
     0 & 1 & 0 \\
     0 & 0 & 0
     \end{vmatrix} \right| = \sqrt{(x-1)^2 + y^2} \cdot \frac{1}{2} \cdot z
     \]

5. **Set up the equations:**
   - Since the quantity is constant, we equate the expressions:
     \[
     \sqrt{x^2 + y^2 + z^2} \cdot \frac{1}{2} = \sqrt{x^2 + (y-1)^2} \cdot \frac{1}{2} \cdot z
     \]
     \[
     \sqrt{x^2 + y^2 + z^2} \cdot \frac{1}{2} = \sqrt{x^2 + y^2} \cdot \frac{1}{2} \cdot z
     \]
     \[
     \sqrt{x^2 + y^2 + z^2} \cdot \frac{1}{2} = \sqrt{(x-1)^2 + y^2} \cdot \frac{1}{2} \cdot z
     \]

6. **Solve for \(z\):**
   - From the first equation:
     \[
     \sqrt{x^2 + y^2 + z^2} = \sqrt{x^2 + (y-1)^2} \cdot z
     \]
     \[
     x^2 + y^2 + z^2 = (x^2 + (y-1)^2) \cdot z^2
     \]
     \[
     x^2 + y^2 + z^2 = x^2 + y^2 - 2y + 1 \cdot z^2
     \]
     \[
     z^2 = 1
     \]
     \[
     z = \pm 1
     \]

The final answer is \(\boxed{1}\).