I was looking at a map of our local stellar neighborhood, and it occured to me, the stars are really close, if one compares them to the size of some nebulae. So can it be, that the Sun, Alpha Centauri system, and all the other stars in the neighborhood actually formed from the same nebula? Have we discovered any information about this?
There are three main reasons why we can tell that local stars did not, for the most part, form from the same molecular cloud that the Sun formed from.
The first is that unless stars are born in a very tightly bound system such as a globular cluster (which the Sun is definitely not in), they will drift apart from their birth companions over time in slightly different orbits in the Milky Way galaxy. The Sun has gone around the center of the galaxy about twenty times since it was born; this is more than enough time for its initial cluster to become completely scrambled in azimuth (the angle in the plane of the galaxy). To see how this would work, imagine two stars with very similar orbits, one with a period of 200 million years and the other with a period of 210 million years. If they start off right next to each other, then after 2 billion years, the first star has made 10 complete orbits, while the second has made about 9.5 -- meaning it will now be on the other side of the galaxy from the first star.
You can also take the current velocities of nearby stars, and see that they can be rather different, which tells you they are not on very similar orbits. Alpha Centauri is moving about 20 km/s in our direction, while Barnard's star has a velocity in our direction of over 100 km/s (in both cases, they also have movement in perpendicular directions; they're not moving directly toward us). So these are stars which happen to be near each other now, but are on different orbits and therefore were almost certainly not near each other at any given time in the past.
The second reason is the one that David Hammen pointed out: stars born from the same clouds should have very similar chemical compositions ("metallicities" in astronomical parlance).
The third reason is that stars born from the same cloud should have the same age (to within a few million years, anyway), since molecular clouds tend to be disrupted by the star-formation process and don't keep forming stars for hundreds of millions or billions of years.
Looking at some of the very nearest stars: the Sun has what we define as "solar metallicity" and is about 4.6 billion years old. The stars of Alpha Centauri are slightly more metal-rich than the Sun and probably several hundred million years older. Barnard's Star is more metal-poor than the Sun (maybe only 1/3 the iron fraction), and about 5 billion years older. Sirius is significantly more metal-rich than the Sun (its iron fraction is about three times that of the Sun) and is only about 250 million years old.
So you can see that the Sun and the nearby stars are a heterogeneous group, and clearly not all born at the same time from the same patch of interstellar gas.
Is it possible that the Sun and all the nearby stars formed from the same nebula?
No, it is not.
Our Sun has marked differences in metals compared to the nearby stars. (In astronomy, every element higher than helium is a "metal".) The discovered and analyzed stars that appear to be most similar to the Sun are far, far away from the Sun. Our Sun apparently formed in what would soon become an open cluster. That means the Sun's sibling stars are spread far and wide. They are not nearby.