The origin is defined as point $O$ at coordinates $(0, 0)$.
A movement of 3 km north leads to point $A$ at $(0, 3)$.
A subsequent turn east and a walk of 4 km results in point $B$ at $(4, 3)$.
A final turn south and a walk of 3 km concludes at point $C$ at $(4, 0)$.
The trajectory begins at $O(0, 0)$ and terminates at $C(4, 0)$.
The straight-line distance between the starting point and the final point is:
\[
OC = \sqrt{(4 - 0)^2 + (0 - 0)^2} = \sqrt{16} = 4 \text{ km}
\]
An alternative perspective involves analyzing the path's net displacements:
The northward movement of 3 km is cancelled by the southward movement of 3 km, resulting in zero net vertical displacement.
The eastward movement of 4 km constitutes the entire net horizontal displacement.
Consequently, the overall displacement is 4 km eastward from the origin.
\[
\boxed{4 \text{ km}}
\]