If everything in existence were increasing in size at some rate, would we be able to detect it? Would our eyes observe any changes?
What about electronic measurement devices?
 A: Depends. 
If you simple assume matter growing we would see the distance between the surfaces of celestial bodies diminishing. Given that we regularly monitor the distance between the surfaces of the Earth and Moon by laser ranging to accuracies of less than one cm (which means less than one part in $10^8$ over the time the project has been running). This is not observed.
If you assume that space is expanding then you have the Hubble expansion and the dark energy, both of which have strong observational support, 
A: Just to add what the other questions say, if the sizes of the atoms were changing, there would have to be some corresponding change in at least one of the fundamental constants.  For instance, if the size of the Hydrogen atom changed, then the ground state of the hydrogen atom would no longer be governed by the Bohr radius:
$$a_{0} = \frac{\hbar}{m_{e}\alpha\,c}$$
which is composed entirely out of fundamental constants of the universe.  You'd have to carefully and subtly change all of these constants to prevent there being any physically measurable effect.
A: You didn't really state what do you imagine when asking your question...
Depending on which level do you want our things to change their size. Here are a few options:


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*The atoms could increase distance between themseves

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*We'd notice this for sure, because things would became larger, but they'd keep their original mass


*All particles (proton, neutron, electron and others) could grow larger

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*I don't think we can detect it. We use particles to watch over core sizes and so all our knowledge of size is relative


*Distances in space

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*We can, of course, detect how far near object in space are - so as for Moon and Sun, yes

*When we talk about objects further away, we actually know they are getting away, thanks to the redshift.


