Is there an object in reality that is proven to be uncountable? I've always wanted to come up with a fairly concrete example of an object or realistic set that could be uncountable. Most of the sets I can think about, even the hugest ones, are always countable. This could be useful to explain to a layperson the difference between countably and uncountably infinite sets.

Can anyone come up with a viable example?
That depends. 

If by "realistic" you mean something that has to do with physical reality, then I defy you to come up with a set which has exactly $200^{200^{200}}$ elements.

If by "realistic" you mean something which comes up naturally in mathematics, then $\Bbb R$ is an uncountable set.

As for explaining the difference between them? That's not very easy, because first you need to be sure that the person understands the difference between sets of size $200^{200^{200}},200^{200^{200}}+1$ and $\aleph_0$. Which is most likely not going to be very easy. Sure, two of them have a finite number of elements, but it's so large it's infinite for all practical purposes. You couldn't even tell them apart if you put the two sets one right next to the other. If they can manage the difference there, then it's not difficult to explain what's "uncountable". Just infinite and not countable.

Unfortunately, mathematicians undergo a difficult training to work with definitions, rather than "common sense intuition" that we have before our studies. So explaining something that had to be earned by hard work is never easy. If it were, we wouldn't have to work so hard to get it.
Set of stars is infinite in reality. Infact it is countably infinite.

Another example of infinite set is "set of all points in a line segment is an infinite set"

See this it says the that set of integers is countably infinite whereas set of real numbers is uncountably infinite.

In countably infinite sets you cannot count the total number of elements but at least you can do the counting. Set of integers is an infinite set but you can count the elements in ascending or descending order(take any starting element). 

In uncountably infinite sets you cannot even count because in between any two real numbers their are infinitely many reals. Here you cannot even do the counting! (suppose you are starting with 0 the you cannot find a number which is just greater then 0, if you think of 0.1 but 0.01 is closer to 0 to get more closer you have 0.001 and so on) this shows that you cannot do the counting.

(Remember you have to do the counting in either ascending or descending order without missing any element in between)   

You can find a formal definition of uncountable sets here. 

You can still compare the cardinality of two sets. If there exists a bijection between two sets then the cardinality of these two sets are equal. The cardinality of $\mathbb R$ and $\mathbb R^+$ is same! (See this)