What is a (naked) singularity?
It is common to think of singularities as being points of infinite curvature on spacetime, or points where gravity is infinite. Using this understanding, it is definitely very difficult to understand how a naked singularity would be a possibility: if gravity is infinite at a singularity, then surely it must be strong enough at a distance from it so as to form a black hole, right?
The thing is: that is not the definition of a singularity. In General Relativity, it doesn't make sense to say "the curvature is infinite at this point", for the curvature tensor is defined in a way such that is must be finite at all points of spacetime. The trick with singularities is that they are not points on spacetime. And this is one of the many reasons General Relativity is hard.
The actual definition of a singularity is roughly the following:
A spacetime is said to be singular if there is an unextendible geodesic that ends at finite affine parameter.
In other words, the spacetime is singular if you can find some observer whose worldline ends at finite proper time (this is actually only a particular case). It is as if you were walking through spacetime with your personal clock and then, at finite time, you suddenly are not at spacetime anymore (no clue where you are or even on whether you exist, but you're not on spacetime). In this sense, singularities are "holes" in spacetime.
Thinking about singularities in this way makes things a bit more clear. One way of creating a "hole" would be to have a region in which curvature is blowing up: the "blow-up point" is then a "hole" in spacetime, a singularity. However, that isn't the only case. As long as you have "holes", you can get singularities. (As a remark, there are naked singularities which are curvature blow-ups. Schwarzschild spacetime with a negative mass is one example).
Intuitively, that is one way of thinking about naked singularities: they are "holes" in spacetime that are not hidden behind a black hole. The issues about these things are that just as an observer can "fall into a hole", you could also have stuff "coming out of the hole". If the singularity is behind a black hole, no one minds, because whatever might come out is trapped inside the black hole and you won't have anything weird. If, on the other hand, the singularity is naked, it means you can't really predict what will happen on the spacetime, because you can't predict what comes out of the hole.
Can one create a naked singularity?
As for whether one can create a naked singularity, that is an open question! As mentioned in another answer, the cosmic censorship conjectures are mathematical conjectures about the structure of General Relativity that state that, under physical conditions, the formation of naked singularities is impossible. Being conjectures, we're still not sure of whether they are true: it could be that they hold and it could be that they do not.
In plain English, the meaning of the two Cosmic Censorship Conjectures is
- Weak Cosmic Censorship Conjecture: under physical assumptions, no naked singularity can arise from gravitational collapse.
- Strong Cosmic Censorship Conjecture: under physical assumptions on the matter and initial conditions of a spacetime, it must hold that it is completely determined by these initial conditions.
The Weak Cosmic Censorship Conjecture is an assumption in many (if not all) of the theorems of Black Hole Thermodynamics, and hence the fact that no mathematical contradictions have been found in Black Hole Thermodynamics suggests that the Conjecture might actually be true. Philosophical reasoning for believing in these conjectures also comes from the belief that the universe should be predictable and causal.
For more details on these themes, you might want to check Wald's General Relativity (especially Secs. 9.1 and 12.1) and his 1993 review paper on the Weak Cosmic Censorship Conjecture.