Vectors can have units. With common vectors like acceleration, velocity, and displacement, the unit of the whole vector is the same as each of its components. That is, a displacement vector of [3 m, 4 m, 5 m] will have a unit of meters. A force vector of [10 N, 5 N] has a unit of newtons. This can help with calculations since, for example, an acceleration vector times mass should yield a force vector.
One way to check that this works is to confirm that the magnitude of a vector has the correct units. The magnitude of a force vector had better have units of newtons. To use the previous example: \begin{align} \left\|\vec{F}\right\| &= \left\|[10 N, 5 N]\right\| \\ &= \sqrt{(10 N)^2 + (5 N)^2} \\ &= \sqrt{125 N^2} \\ &\approx 11.2 N \end{align} as expected.
In other areas of physics, vector components can have different units. In optical ray tracing, the components of a ray vector are a distance and an angle. So, the vector does not have an overall unit. One consequence is that, unlike the force vector example above, the magnitude of a ray vector is meaningless.