In a closed economy, savings equal investment. Your equation actually shows this. We have:
$Y−(C+G) - I = 0$.
Note that savings, by definition, is just equal to the production $Y$ that is not consumed. Here we have consumption by private citizens $C$ and consumption by the Government $G$. Therefore, savings $S$ is:
$S = Y - (C+G)$
So national savings can only be zero if investment is zero.
Here is a concrete example. You don't spend all your income and save money in a bank. People who want to invest (e.g. to start a business or buy a tractor) go and borrow that money, which is then invested.
In an open economy, there is no reason for savings to equal investment. Nevertheless, the two are often strongly correlated in reality, which is known as the Feldstein-Horioka Puzzle. This is/was one of the major puzzles in international economics and a large literature has been devoted to explaing it.