Is it possible to determine the absolute value of entropy of a system?
Well, in practical I would say it is not possible. But there is a way to theoretically approximatly calculate an absolute value for entropy. 

According to the third law of thermodynamics the entropy change can be calculated as 

$$\mathrm d S = \frac{\delta q}{T}\, .$$

Per definition, a perfect crystal at $\pu{0 K}$ would have an entropy $S=0$.

So when you now heat that crystal and measure the change in entropy, since your initial $S$ was zero, the change is equal to the absolute value.

One problem is though, that the above equation only holds for constant temperatures. So the steps at which you measure the entropy changes have to happen at very (infinitesimal) small temperature changes.