Whenever I hear someone describe quantum physics as "weird" - whenever I hear someone bewailing the mysterious effects of observation on the observed, or the bizarre existence of nonlocal correlations, or the incredible impossibility of knowing position and momentum at the same time - then I think to myself: This person will never understand physics no matter how many books they read.
Reality has been around since long before you showed up. Don't go calling it nasty names like "bizarre" or "incredible". The universe was propagating complex amplitudes through configuration space for ten billion years before life ever emerged on Earth. Quantum physics is not "weird". You are weird. You have the absolutely bizarre idea that reality ought to consist of little billiard balls bopping around, when in fact reality is a perfectly normal cloud of complex amplitude in configuration space. This is your problem, not reality's, and you are the one who needs to change.
Human intuitions were produced by evolution and evolution is a hack. The same optimization process that built your retina backward and then routed the optic cable through your field of vision, also designed your visual system to process persistent objects bouncing around in 3 spatial dimensions because that's what it took to chase down tigers. But "tigers" are leaky surface generalizations - tigers came into existence gradually over evolutionary time, and they are not all absolutely similar to each other. When you go down to the fundamental level, the level on which the laws are stable, global, and exception-free, there aren't any tigers. In fact there aren't any persistent objects bouncing around in 3 spatial dimensions. Deal with it.
Calling reality "weird" keeps you inside a viewpoint already proven erroneous. Probability theory tells us that surprise is the measure of a poor hypothesis; if a model is consistently stupid - consistently hits on events the model assigns tiny probabilities - then it's time to discard that model. A good model makes reality look normal, not weird; a good model assigns high probability to that which is actually the case. Intuition is only a model by another name: poor intuitions are shocked by reality, good intuitions make reality feel natural. You want to reshape your intuitions so that the universe looks normal. You want to think like reality.
This end state cannot be forced. It is pointless to pretend that quantum physics feels natural to you when in fact it feels strange. This is merely denying your confusion, not becoming less confused. But it will also hinder you to keep thinking How bizarre! Spending emotional energy on incredulity wastes time you could be using to update. It repeatedly throws you back into the frame of the old, wrong viewpoint. It feeds your sense of righteous indignation at reality daring to contradict you.
The principle extends beyond physics. Have you ever caught yourself saying something like, "I just don't understand how a PhD physicist can believe in astrology?" Well, if you literally don't understand, this indicates a problem with your model of human psychology. Perhaps you are indignant - you wish to express strong moral disapproval. But if you literally don't understand, then your indignation is stopping you from coming to terms with reality. It shouldn't be hard to imagine how a PhD physicist ends up believing in astrology. People compartmentalize, enough said.
I now try to avoid using the English idiom "I just don't understand how..." to express indignation. If I genuinely don't understand how, then my model is being surprised by the facts, and I should discard it and find a better model.
Surprise exists in the map, not in the territory. There are no surprising facts, only models that are surprised by facts. Likewise for facts called such nasty names as "bizarre", "incredible", "unbelievable", "unexpected", "strange", "anomalous", or "weird". When you find yourself tempted by such labels, it may be wise to check if the alleged fact is really factual. But if the fact checks out, then the problem isn't the fact, it's you.
as if it made sense to say of a particle that it has a position, but no particular position
That might or might not make sense (mathematicians have been tearing their hair out about non-computable numbers, see Chaitin's constant). But most quantum mechanists do not say that a particle has a position. In fact if you interpret Quantum mechanics in terms of "hidden variables" (there are underlying values for the objects, like spin and momentum, but we can't get at them) then you will generally come unstuck.
Can you explain to me the exact nature of this 'combination' that is the actual property?
The property is exactly the one in the quantum formalism. I don't really see why you object to the formalism. It gives specific predictions that have been confirmed, with high probability, in experiments.
If you want an ontological view, then my position is that science is only about making predictions about the results of experiments and then testing them. Properties such as position, energy, etc... are only valid in that they predict a lot of different experiments. In classical mechanics, it emerged that a mathematical concept called "position" led to great predictive power, giving universal laws. So classically, "position" existed.
In quantum mechanics, laws based on "position" don't work, so the concept of position doesn't make sense in a quantum framework (just as "colour" makes no sense in acoustics). Other concepts did make sense - they had to be expressed in certain formal mathematical ways, but they made sense.
So, to sum up, position doesn't exist, momentum doesn't exist, but certain other objects (such as the product of the uncertainties of momentum and position) do make sense.
Aha! But have I not defined "uncertainty of position"? How can I claim this exists if position doesn't? The problem is just the name (and this is going back to Elizer's original point, and causing me to think I may have been a bit hasty in rejecting it). This is just the standard deviation of an observable. It's only called "uncertainty of position" because of an analogy with the classical "position" - a wrong analogy (and an observable, like a classical "position", is just a mathematical object that seems to make sense in experiments).