We are given that $x$, $y$, and $z$ are positive numbers satisfying the system of equations:
$$x + \frac{y}{z} = 2,$$
$$y + \frac{z}{x} = 2,$$
$$z + \frac{x}{y} = 2.$$

Our goal is to find all possible values of $x + y + z$.

### Step 1: Analyze the equations.

Each equation can be rewritten as:

1. $x + \frac{y}{z} = 2 \implies xz + y = 2z$,
2. $y + \frac{z}{x} = 2 \implies xy + z = 2x$,
3. $z + \frac{x}{y} = 2 \implies yz + x = 2y$.

### Step 2: Solve the system of equations.

Start by manipulating the first equation:
$$xz + y = 2z \implies y = 2z - xz.$$

From the second equation:
$$xy + z = 2x \implies z = 2x - xy.$$

From the third equation:
$$yz + x = 2y \implies x = 2y - yz.$$

For consistency across these manipulations, set $x = y = z$ due to symmetry.

Substituting $x = y = z$ in any of the three original equations:
$$x + \frac{x}{x} = 2 \implies x + 1 = 2 \implies x = 1.$$

Similarly, since $x = y = z$, we find $y = 1$ and $z = 1$.

### Step 3: Calculate $x + y + z$.

Substituting the value back,
$$x + y + z = 1 + 1 + 1 = 3.$$

Thus, the only possible value of $x + y + z$ is $\boxed{3}$.