In any situation where the electromagnetic force is present, it’ll always dominate over the weak force, until you get up into energies of nearly a hundred GeV. The strong force will eventually become repulsive and overcome the electromagnetic force, but we don’t know the details of precisely how the strong force works.
“Fit in one cup”? No. You’re presumably packing spheres here, so you’re looking at a best-case density of .75
The answer you’ve given is the number of protons that will take up the space of one cup. But you couldn’t fit that many INTO a cup.
2 protons, one cup.
snerk
Thanks for the answers. I guess I hadn’t really thought about the logistics of getting that many protons to stay that close together. Would this thought experiment be any easier if we used neutrons?
Actually – for any given ambient pressure – protons sans electrons will be less dense, as they’ll repel each other. Electrons will soak up the charge and allow closer packing.
yeah, the protons would all “burn” away leaving nothing but a cup of Neutrons - hence the term “Neutron Star”
See, this is why they sell protons by weight, not by volume. Settling or expansion of contents may occur during shipping.
At this point, have we not established that this is a thought experiment, and that there are many many real world constraints that would prevent anyone from filling up a cup with protons?
The intent of the OP is pretty clear, and calling it impossible to do is true, but valueless. This question as posed is akin to “What would happen if you cut a hole through the earth and jumped down” or “What if you turned your headlights on when you were going at c” – both impossible experiements but conceptually useful. If the OP wants to clarify his question to ask something like the maximum density of proton matter achievable, then you can remind everyone that like charges repel.
The answer to the question posed is “an immense amount”.
Answering the question requires filling in all sorts of unknown assumptions. The simplest assumption is the weight of one cup of hydrogen gas. You’d assume that would be a sea level temperature and pressure, I suppose. The weight of the electrons would be trivial. But then we find an interesting thing, when we put the protons (+ a tiny weight of electrons) on a scale to see how much they weigh, they float away. Negative weight! So should we change the question to, “How much do they mass?”
We can’t just say, “this is a thought experiment”, because there’s no answer to the question without making about 30 assumptions not given in the question. It’s like asking, “How long is a piece of string?”
Unless otherwise specified the conditions are standard temperature and pressure. Basically sea level pressure and room temperature. The same assumptions you would make for any other liquid or gas. So a cup of protons would not weigh very much because to confine a lot of them will require a large pressure due to the fact that they repel each other.
This, along with Flymaster’s correction, seems to me to be best answer. When someone asks ‘what would a cup of X weigh’? I would assume that they mean if it were all packed in tight.
The other answers seem to suffer from an over-abundance of cleverness.
But if someone asks, “What would a cup of lead weigh?” or “What would a cup of yogurt weigh?”, we can make pretty good assumptions about what it means to be “packed in tight”. Even for “a cup of air”, we could imagine sea level pressure and room temperature.
But a cup of protons can’t be “packed in tight” under sea level pressure and room temperature. To get an answer for protons “packed in tight”, we’d have to increase the pressure to a point where we’re not talking about ordinary matter. Or we could be talking about a cup full of very thin ionized hydrogen gas.
And if the shipping is really slow there may be some spoilage 
Has science determined the shelf life of protons yet? I seem to remember big experiments trying to determine the half life of protons. A few years ago they had not yet seen a proton decay.
The string-length is approximately the Planck length, but can be significantly bigger when the strings are weakly interacting. The plank length varies - a standard 2x4 comes in many different lengths.
A proton, on the other hand, would not fit in the other hand; nor in a cup. However they do have cup holders.