Is uncertainty a physical obstacle? Heisenberg's Uncertainty Principle states that you cannot know the position and the momentum of a particle at the same time (I believe this is the main idea behind it). And I have read in various places an example of that which goes something like this:
If you were to look at an electron under a microscope, you have to have a photon bouncing off of it and then travelling to your eye in order to see it. But this mere act of a photon hitting the electron changes its momentum so by the time you know about its position, the momentum is changed so you can't know them both at the same time. 
But this seems like a physical problem to me. What if we could find out the position of an electron without ever interacting with it physically? Would the momentum then be unknown to us too?
 A: The Heisenberg uncertainty principle is tied up with the whole formalism of quantum mechanics. The formalism of quantum mechanics, differential equations and postulates, have never been falsified experimentally up to now, whereas there exist innumerable validations of the theory. 
What you were given as a description is misleading and leads to confusions

It has since become clear, however, that the uncertainty principle is inherent in the properties of all wave-like systems,[4] and that it arises in quantum mechanics simply due to the matter wave nature of all quantum objects. Thus, the uncertainty principle actually states a fundamental property of quantum systems, and is not a statement about the observational success of current technology.

The basic equations of QM are wave equations, but not of the usual classical waves where energy varies in space and time. The interpretation of the solutions of the QM equations depend on the postulate that "the square of the wavefunction gives the probability distribution for measuring a given observable". i.e. the probability distribution measured by many single event measurements of the given observable, statistically, is given by the square of the wave function which is the solution of the QM wave equation ( Schrodinger, Klein-Gordon, Dirac) for the given boundary conditions of the problem/experiment.
The word "probability" itself carries uncertainty.
A second  postulate of QM is " to every observable there corresponds an operator". These are differentials of some form on the wave function, by  space and time variables.   In addition, the mathematical construction of the QM equations in terms of operators, i.e. differentiations, introduces pairs of observables that do not commute, i.e. when operating on the wavefunction to get the observables, AB is not equal to BA: this link gives a good outline of how these commutators work. The most well know is the [x,p] commutator and  it gives us the standard Heisenberg uncertainty between momentum and position.
So in conclusion, all our data confirm the quantum mechanical view of the microworld  and the mathematical theory leads to the uncertainty principles rigorously. No , the HUP is not artifact of measurement but a fact of how nature behaves in the microworld. 
