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?

In some ways this may be a philosophical question.
My view is that the randomness a statistician sees in
data analysis is often real. Randomness might result from random sampling or from inherent randomness or instability.
Either way, statistical procedures of estimation might be useful.
Let's look at two
entirely different situations.
(1) We wonder whether the true average height of women high school
seniors in a US state is 65 inches. At noon on a particular day, one might carefully measure each
of them (thousands or hundreds of thousands, depending on the state), round to the nearest tenth of an inch, and take the mean.
Even
with perfectly accurate measuring and record keeping,
the result would likely be different on another day. Some students
might have grown just a bit. Also, there is some evidence that
a person's height may depend (slightly) on how much sleep they got the previous night. Does it make sense to say that the true statewide average height fluctuates constantly in some random way?
More realistically, one might take a random sample of $n = 400$ women for the measurements and averaging. In this case,
random sampling has clearly induced randomness. It would not
be surprising if another random sample of $400$ resulted in a different mean. Using such a random sample we could
make a 95% confidence interval for heights of senior women.
Perhaps this CI would be about $0.7$ inches wide, but we
could be fairly sure that the interval contains the right
answer for the heights of senior women in the state. For
practical purposes that might be sufficiently close to the 'correct' answer. Perhaps some probability model such as
$\mathsf{Norm}(\mu = 65, \sigma=3.5)$ is approximately correct.
If so, then our two samples of size $400$ might (at random) have been
$65.2$ and $65.0.$
round(rnorm(2, 65, 3.5/20) ,1)
[1] 65.2 65.0

(2) You have a small specimen of some radioactive material
with a long half-life (several thousand years). You use an appropriate counter to
count particles emitted in one particular minute to be $746.$
Using exactly the same experimental set-up several minutes later the count is $721.$
Monitoring the specimen over
several hours, you believe the hourly rate of particle emission
in your experiment is distributed $\mathsf{Pois}(\lambda=750).$ As far as is known, the number of particles
emitted into our counter per minute is truly random, and $\mathsf{Pois}(750)$ is a reasonable model. Both of our one-minute measurements could be exactly correct, but it would be wrong to
expect them to be exactly the same. (But not often fewer than 697 or more than 804.)
set.seed(1226); rpois(2, 750)  
[1] 746 721
qpois(c(.025,.975), 750)
[1] 697 804

