[infool7]

Is it the Coulomb Barrier that prevents the nuclei from actually touching?

If so then it’s probably a good thing.

Don’t touch it, don’t talk about touching it, don’t write songs about touching it!

[Boogieman]

Initially, but once you get past that, there is still the Pauli Exclusion Principle.

[infool7]

Found the movie quote from Walk the line:

Jerry Lee Lewis: God gave us a great big apple, see, and He said don’t touch it. He didn’t say touch it once in a while; He didn’t say take a nibble when you’re hungry; He said don’t touch it! Don’t think about touchin’it, don’t sing about touchin’ it, don’t *think* about singin’ about touchin’ it!

[expat2]

Mostly -- it prevents them getting too close unless there is a lot of energy. A neutron can be attached to the proton relaatively easily, to form deuterium, which is stable.
The helium atom has two electrons, with two protons and two neutrons in its nucleus. (The strong force overcomes the coulomb force when the protons and neutrons are 'inside' the coulomb barrier.) That nucleus is an alpha particle if the 2 electrons are missing (He2+).