There are $2017$ lines in the plane such that no three of them go through the same point. Turbo the snail sits on a point on exactly one of the lines and starts sliding along the lines in the following fashion: she moves on a given line until she reaches an intersection of two lines. At the intersection, she follows her journey on the other line turning left or right, alternating her choice at each intersection point she reaches. She can only change direction at an intersection point. Can there exist a line segment through which she passes in both directions during her journey?
1. **Assume for contradiction** that Turbo the snail can pass through a line segment in both directions during her journey. This implies that Turbo must visit the same intersection point twice.

2. **Consider the path** Turbo takes: $v_1 \rightarrow v_2 \rightarrow \ldots \rightarrow v_n \rightarrow v_1$, where $v_i$ is an intersection point of two lines. 

3. **Note that on each segment** $v_i v_{i+1}$ (with $v_{n+1} \equiv v_1$), there are no other intersection points except $v_i$ and $v_{i+1}$. This is because no three lines intersect at the same point, and Turbo only changes direction at intersection points.

4. **Observe that each line** containing a segment $v_i v_{i+1}$ does not intersect any other segment of the path. This implies that the path $v_1 v_2 \ldots v_n$ forms a simple polygon.

5. **Since the path forms a simple polygon**, and given that Turbo alternates her direction at each intersection, the path must be a convex polygon. This is because in a convex polygon, each internal angle is less than $180^\circ$, and Turbo's alternating direction ensures that she cannot revisit any intersection point without violating the convexity.

6. **However, this leads to a contradiction**. If Turbo were to revisit an intersection point, the path would no longer be a simple polygon, as it would have to cross itself, which is impossible given the initial conditions (no three lines intersect at the same point).

7. **Therefore, Turbo cannot pass through the same line segment in both directions** during her journey.

\[
\boxed{\text{False}}
\]