Is the Portal feasible in real life? Firstly, I assume prior knowledge/experience with the game Portal.
Should it be possible that two openings (circles/unidentified-constructions) would work as interchangeable gates? With one opening analyzing the molecular structure of the object passing through and the other reconstructing it accordingly? Maybe as a replica?
The object doesn't need to be alive or biological at all.
 A: The game acknowledges within the first 30 minutes of the game that they violate the law of conservation of momentum. GLaDOS says something along the lines of
"Now you will learn your first lecture about conservation of momentum. Or rather, the lack thereof."
http://lshs.leesummit.k12.mo.us/studentprojects/1011/fall/hr1/bmckenna/html_project1/images/Portal%20Diagram.png
A feature of the game is that the portal seems to conserve the absolute value of momentum or speed, but not the vector direction $\vec p$. Going through the portal in the following picture...

(source: thebuzzmedia.com) 
...would correspond to $\vec p \rightarrow -\vec p = \vec p-2\vec p$, which would be like getting pushed by $2\vec p$.
But I'm making a not here: Even if it's not in the object-transportation sense of the game and although the seperation of the two points is not as arbitrary, you might wanna read up on Quantum Teleportation. There is still research to be done.
Lastly, the problem is that you can't really make statements about an object, where nobody knows how to construct it. So this is really a sci-fi question and therefore not suited for physicsSE. 
On the other hand, I always wanted to hear some elaborations on the topology of a world with portals:

