Other people have answered why the probability is zero (if you approximate time as being continuous, which it is effectively not, but anyway...) so I will just echo it briefly. To answer the last question that the OP asked---"how could it occur if it had probability 0?"---lots and lots of things can occur if they have probability zero. All a set of probability zero $A$ means is that, in the space of possible things that could happen, the set $A$ takes up no space. That is all. It is not more meaningful than this.
I am writing this to hopefully address something else that the OP said in the comments:
You say "you will never hit the point zero", but what can you say of the point that I hit in my first dart throw? Let 𝑥 be the point that I hit. Before throwing my dart, you would have said "you will never hit the point 𝑥", but I've just hit it. Now what?
This is a very good question and one that, when I began to learn about probability, I struggled with. Here is the answer: it isn't equivalent to the question that you originally asked! What you have done is bring time into the analysis, and that means that the underlying probability structure changes to become much more intricate. Here is what you need to know. A probability space $(\Omega, A, \mu)$ consists of three things:
an underlying space $\Omega$, such as $\mathbb{R}$ or $\mathbb{Z}$; a set of all possible outcomes on this space, such as the set of all half-open intervals on $\mathbb{R}$, and a measure $\mu$ that satisfies $\mu(\Omega) = 1$.
Your original problem lives in the space
$([a,b], \text{all half open intervals on $[a,b]$}, \nu)$ where
$\nu$ is Lebesgue measure (this means that $\nu( [c,d) ) = \frac{1}{d - c}$). In this space, the probability that you hit any single point $x \in [a,b]$ is zero for the reasons discussed above---I think we have this cleared up. But now, when you say things like the quoted passage above, you are defining something called a filtration, which we will write as $\mathcal{F} = \{\mathcal{F}_t\}_{t \geq 0}$. A filtration in general is a collection of subsets of $A$ that satisfy $\mathcal{F}_t \subseteq \mathcal{F}_s$ for all $t < s$.
In your case, we can define the filtration
$$
\mathcal{F}_t = \{x \in [a,b]: \text{dart hit $x$ at time $t' < t$} \}.
$$
Now, in this new subset of your outcome space, guess what---you're right! You have hit it and, after your first throw, your probability of having hit that point when restricted to the filtration $\mathcal{F}_1$ is 1.