Is anything inherently random? Is anything inherently random? Or is all randomness observed in data either "errors in measurement" or "lack of understanding"? Assume we could measure everything with infinite precision and had a complete understanding of all the deterministic processes. Would there be any process that would appear "random" in the data or would every dataset be deterministic at that point?

We do not know and may never know.  (and possibly we can not know*)
Statistics is not about the nature of reality. Statistics describes observations and these observations happen to have a random appearance due to unknown variable factors that are involved in the model descriptions.
The question whether the reality is intrinsically non-deterministic is more like philosophy than statistics.

*One may argue that the question about the deterministic nature of the world is actually not something that can be answered by a consciousness within this reality, and it may even not have any meaning at all. If the physical processes occur with some random nature, that means that the future state of the world can not be determined based on the present state, then still something makes the current state evolve into the future state and it is in some way determined. It is only not determined by an entity that can be known by the objects in the observable world. Whether or not something is deterministic is a matter of perspective.
