Pick a number between $0$ and $1$ inclusive. What is the chance you pick $0$?
Well, the chance that you pick something less than $.5$ is $1/2$. Likewise, the chance you pick something less than $.25$ is $1/4$, etc. Continuing this logic, the probability you pick $0$ must be less than any positive number. But negative probability doesn't make sense, so the answer must be that the probability of picking $0$ is exactly $0$.
You can make a similar argument and show that the probability of picking any number in the range $[0,1]$ is exactly $0$.
If you look past $[0,1]$ and allow picking any real number, the answer remains exactly the same for similar reasons.
What if you picked $n$ random numbers? Would it make any difference? No - $n\times0 $ doesn't make your odds any better.
What if you picked an infinite number of numbers? It turns out that any list $x_1, x_2, x_3, \ldots$ is, in a sense, infinitely smaller than the set of all real numbers, or even the set $[0,1]$. For example, I could start listing all of the rational numbers in $[0,1]$:
and I would hit any rational value less than one eventually. You cannot list all of the real numbers in this way, you will always leave infinitely many out of your list. You can read about this more here.
The probability of picking a number from $[0,1]$ and getting a rational number is $0$, and any infinite list is in one-to-one correspondence with the rational numbers. Because of this, you have $0$ probability of picking the number in question even with an infinite list of guesses.
How can it be that you have $0$ probability of doing something that is clearly possible? Say you pick a number $x$ randomly from $[0,1]$. How could you have done that if the chance of picking $x$ was zero to start with? It turns out this is not inconsistent, probability in infinite spaces is just counter-intuitive. Mathematicians use the term almost certainly to describe things that happen with probability $1$.