This actually gets even worse. Consider for example a hypothetical Bayesian version of Issac Newton, trying to estimate what exponent k the radius is raised to in F= GMm/R^k. There's an intuition that mathematically simple numbers should be more likely, such as say "2". A while ago jimrandomh and benelliiot discussed this with me. Ben suggested that in this sort of context you might just have a complicated distribution where part of the distribution arose from something continuous and the other part arose from discrete probabilities for simple numbers. This seems to do a decent job capturing our intuition but it seems to be very hard to actually use that sort of distribution.

It may or may not be helpful to realize that infinities (including infinitesimals) are merely a mathematical abstraction. Everything you encounter in the physical world is finite. Thus, it's not overly surprising that something actually happens, even though a given mathematical model of that something assigns it a zero probability.

That said, mathematical descriptions that include continuity are extremely convenient (life would be rather cumbersome if we had to use finite difference calculus instead of derivatives in all applications).

It is a very common tendency to identify a physical phenomenon with a particular mathematical model of it (one of the most abused models is that of virtual particles in particle physics), but one would be rather less wrong by keeping in mind that an abstraction of an object is not the object itself.

A nice (if fantastical) description of objects vs models can be found in the HPMoR chapter on partial transfiguration.

Let X be a random variable over the interval [0, 10]. Then, by the definition of probability over continuous domains, P(X = 1) = 0.

Only if you have a continuous probability distribution over that domain. It's quite possible to have a probability distribution with, for example, a point mass at 5 such that p(X=5)=0.5.

This is sometimes described as counterintuitive: surely, at any measurement, X must be equal to something, and thus its probability cannot be zero since its clearly happened.

Others have answered this below, but there is another aspect to this I'd like to discuss. All data are discrete. When you measure something, your measurement apparatus is only ever going to give you one of a discrete, finite set of values. (I'm pretty sure about finite, but willing to be corrected). Any probability distribution over the possible values that you might measure with your apparatus can easily satisfy p(X=x)>0 for all x.

Concretely, if you're measuring the length of something with a ruler, you probably just round to the nearest 1/16th of an inch. This means there are only 12*16=192 possible measurements you can make, so you can create any number of probability distributions of these points where each point has p(X=x)>0.

You are misunderstanding what probability means. A probability of 0 does not mean an event will never happen, it means it will almost surely not happen. That is, for any finite possibility one can think of, the probability of that event occuring is less than that. This does not mean that the event can never occur- as you surmise, otherwise we could never observe any result! Basically, infinity is a bit weird.

My intuition, for what it's worth, works more easily with the binary expansion of the numbers than their interpretation as physical quantities.

From that perspective, "X equals exactly pi" would normally be assigned finite probability because pi is a computable number with finite Kolmogorov complexity; there is a nonzero chance that two processes will generate the same infinite but computable bit stream.

But "X equals exactly Y" where Y is a random incomputable number, is indeed infinitely improbable, because it amounts to a statement that infinitely many coin flips will come out a particular way; the probability is 0.5^infinity, which clearly converges to zero.

An uniform distribution over the real interval e.g. [0,1] is possible. An algorithm is the fair coin tossing for each binary place 0 or 1. In the case of all tails you have 0. In the case of all heads - it's 1. An uniform probability distribution with P(x)=0 for every x. They are not impossible, only 0 likely.

But there is no constant probability distribution for only the rational numbers from this interval. Or from any other, for that matter. Nor there is an uniform probability distribution for all the naturals. Or for any infinite subset of naturals.

Then of course there is quantum physics, where it is literally impossible for any physical object, including point objects, to have a definite coordinate with arbitrary precision.

Conventionally, in the multiverse, everything is precisely somewhere. What is difficult is finding out exactly where things are.

Given that the sum (0+0+0...) = 0, wouldn't that imply that P(any value at all) = 0, and that you actually cannot produce a result in this system?

Which, admittedly, strikes me as a perfectly reasonable result, given you can't actually have a continuous distribution in reality, and I'm not aware of any randomization method that could actually meet these requirements.

P(X = exact value) = 0: Is it really counterintuitive?

Not counterintuitive, just annoying. Also used as an excuse to conclude silly things by playing with infinities (as you allude to).