What does it mean for the total energy of a closed system to be zero at all times? If a closed system has kinetic and potential energy such as the total energy (the sum of the two) equals zero for all times, what does that mean? In other words, what does it physically mean that the total energy is always zero for a closed system?
I think I have a small misunderstanding of the interpretation because i ask myself: how can the system do anything at all if its total energy is zero? But at the same time i think, one can choose the zero potential energy such as the total energy is zero.
 A: It doesn't mean much that the total energy is zero - although in practice if this happens it's likely that the system's potential energy is negative, which is balancing the positive kinetic energy.
Let's take the Earth as an example. By convention, the potential energy is set to zero at infinitely far away. Because gravity is attractive, you don't have to exert force on a faraway object for it to start falling towards the Earth. Therefore, the potential energy of a body on the surface of the Earth is negative (given by $PE = - GMm/r$). If the kinetic energy of the body is exactly equal to $GMm/r$, then the Earth-body system would have zero total energy. The body is still moving though, since it has a nonzero kinetic energy. You can also extract more potential energy from the body if it's able to fall deeper into the Earth (e.g. you could dig a hole and drop the body in), just like you are able to convert some energy into potential energy by raising the body higher up.
Ultimately it's not the exact number you get for the energy that matters, but the change in energy, which is related to the total work done.
A: Because you can always add a constant to the potential energy without changing the physics. In other words, if you have $$E=K+U=0$$ you will not change anything if you move the zero-point of the potential energy so that $$E=K+U+U_0\neq0$$
Additionally, while $K$ is always non-negative, we can always move to a different inertial frame that has a different speed relative to our previous inertial frame, and so even $K$ is not an absolute value for a system. 
Therefore, really a better question is

What does it mean for the total energy of a closed system to be constant?

When total energy is constant, then you can easily relate change in potential energy to change in kinetic energy, as
$$\Delta E=\Delta K+\Delta U=0\to \Delta K=-\Delta U$$
In other words, constant total energy means that any loss / gain in kinetic energy comes from a gain / loss in potential energy.

How can the system do anything at all if its total energy is zero?

This isn't much of an issue once you realize that total energy can be negative (because potential energy can be negative). Then $E=0$ is not an indication of "the system has no energy to give". 
