Suppose we are picking points uniformly at random from the surface of the Earth. I want to compute the probability that I pick a point in the Western hemisphere, given that I pick a point on the equator. The answer should clearly be $1/2$.
From the definition, we have
$$P(A|B)=\frac{P(A\cap B)}{P(B)},$$
where event $A$ is choosing a point in the Western hemisphere and $B$ is choosing a point on the equator. As the equator is a $1$-dimensional smooth line on a $2$-dimensional surface, it has measure $0$. So I compute $P(B)=0$. But using the conditional probability formula requires $P(B)>0$. In fact, this is the definition of conditional probability! So how do I make sense of $P(A|B)$? Clearly, it should work out to be $1/2$, but what is the rigorous way to compute it?
This is a surprisingly philosophical question, and as such, here is a link to a philosophical paper about it: What Conditional Probability Could Not Be
Practically speaking though, you're absolutely correct - this probability is $1/2.$ However it is difficult to describe this fact using conditional probability the way it is usually understood.
The way I would "rigorously" approach this problem the following: let's say you have a probability space $(X,\Sigma,P)$ and a subspace $Y\subset X$ such that $P(Y)=0$. How do we 'condition' on this space? Well, the same way we consider a "line" integral in $\mathbb{R}^2$: $Y$ becomes your new universe, so you have to define a new probability space $(Y,\Sigma_2,P_2)$ where you can answer questions such as this. The statement $P(A\cap B)/P(B)$ is somewhat like trying to measure the length of a line segment using a bathroom scale - the scale ignores the line segment, so you have to get a ruler instead!