What are the borders of determinism? I, a newbie in physics, often read about "near determinism", which is most probably the actual state of physics, meaning: the "big world" is deterministic, but very small things (atoms and smaller) are indeterministic (e.g. quantum physics). 
If this is true, where is the border?
At which size do objects in our universe stop to be indeterministic and start to be deterministic?
 A: There is a relationship that determines indeterminism  :).  It is called the Heisenberg Uncertainty Principle.
Size can be described by the variable $x$ for the position of a particle/atom/molecule. The principle says that we can only know the value of $x+\Delta x$ and  the momentum of the particle  $p+\Delta p$ (where $\Delta x,\Delta p$ denote small intervals) within a relationship bound by
$$\Delta x\cdot\Delta p \gt  \hbar/2$$
where $\hbar$ is the reduced Planck constant
This means that if we want great accuracy in position the momentum will be indeterminate. 
Equally if we want great accuracy in Energy, Time will be indeterminate.
From the wiki link:

the uncertainty principle actually states a fundamental property of quantum systems, and is not a statement about the observational success of current technology.It must be emphasized that measurement does not mean only a process in which a physicist-observer takes part, but rather any interaction between classical and quantum objects regardless of any observer.

There is no unique border, it depends on the variables under observation, but 
ħ  is a very small number, which can be approximated with 0 in the macroscopic world. The Heisenberg uncertainty is relevant for the mircoscopic world of atoms and molecules and smaller.
A: The border is fuzzy. 
The border is roughly determined by the value of Plank constant. If the values of the task is close to it, then quantum mechanics guides the scene.
More explicitly, atom parts (like electron orbitals) are mostly in-deterministic, while molecules, including DNA molecules, are mostly deterministic.
A: I'd agree with Ali that the answer depends on the interpretation. For example, the Bohm interpretation (http://en.wikipedia.org/wiki/De_Broglie%E2%80%93Bohm_theory ) is a striking example of a deterministic interpretation.
