Definition of Open set.

... then you take a smaller radius.

A set $Z$ is open if for every point $x$ you can find an open ball (which might be very small) centered at $x$ that is contained in $Z$.

It doesn't mean every ball will work. It only means some balls will work. Actually it only needs to mean at least one ball will work. [That usually means the smaller balls will work too but sometimes there aren't any smaller balls (don't worry about that now.)]

If for example if you take the interior of a unit circle in a plane centered at $(0,0)$. That's open because "it has fuzzy edges"... Anyway if you take any point in it. Say $(0, 0.9999)$ then I can find an open ball centered at $(0,0.9999)$ entirely in the circle. To do that I have to take a radius $r \le 0.0001$; but I can do it if I take a radius small enough.

Now what if instead I had taken a radius of $27$ and that's way too big to fit in the circle? Well, I shrug my shoulders and give the person asking a confused look and say "Who cares about that radius? I can take a radius of $0.0001$ and that does work. I dont care about radii that don't work. I just care that there are radii the do work."