Welcome to PF Uncertainty.
First some practical info... to measure something on scale x, you need something which is of that scale. For example, when you look through a microscope, you are sending light at an object and looking at the reflection (optically magnified). Light has a wave-length of several nanometers (visible light is approximately 300 - 800 nm). Light can reflect of things which are larger than this wavelength, but when the object is smaller the light will not reflect (an intuitive, but probably wrong, picture I have is that the wave is too stretched to actually hit the object and will just pass it). So with microscopes we can only see objects to about 300 nm. If we use, for example, X-rays, we can go a bit further. Quantummechanically, however, all particles have a wavelength (because they are simultaneously waves) called the de Broglie wavelength. The wavelength of electrons is still much smaller than that of electromagnetic radiation such as light or X-rays. This is being applied in electron microscopes, where instead of light (photons), electrons are shot at a target (and processed by a computer to produce an image), allowing us to see much smaller scales and indeed resolve separate atoms.
So currently, the scale to which we can zoom is limited by technological factors. If we want to see smaller scales, we will need smaller objects to probe them. Compare it to this: if you shoot tennis balls at a wall and look how it comes back to you, you only get rough information about the surface of the wall. But if you shoot ping pong balls, they will also recoil from smaller cracks in the surface and you will also get information about a finer structure.
Now, as for the theoretical part.
As far as we know now, we can measure positions to arbitrary precision. However, there is a quantummechanical "uncertainty" principle, named after Heisenberg, which tells us that we have to pay a price for such precision. For example: the better we know the position of a particle, the less we know about its momentum (velocity). This has to do - again - with particles actually being waves in quantummechanics.
We think that all our current theories break down at some point. At very small length scales, very high energies and very large momenta, the theory we have will stop being valid, in much the same way as classical mechanics makes way to special relativity for large momenta and for quantum mechanics at small length scales (and quantum field theory at large momenta and small length scales, at the same time).
Currently theoretical physicists are speculating what would lie beyond this point. One general idea is that below a certain scale (called Planck scale) space(time) will start looking granular. Then arbitrary measurements are no longer possible, we can only measure distances in terms of some fundamental distance for example. This is comparable to electric charge, which can only be measured in terms of some fundamental charge e (electron charge). However, the Planck scale lies far beyond anything we can measure today, or even in the near future. Maybe we will never be able to measure it at all. Therefore the greatest challenge for such theories is to postulate some microscopical structure (say, on the Planck scale) of space and try to average out these effects to predict something on a scale which can be reached now or in the near future.
