This can be done by a slightly "hacky" way WITHIN Mathematica. You may find the data info behind the graphics obj by hitting shift+ctrl+E (on mac: use cmd instead of ctrl) after you choose the graphics. Please follow the short example: Now you should have a plot stored in the variable "plt". Then use the Part function ([[ ]]) to extract the information from the data list, which is “plt“ as well. The actual index may vary depends upon the graphics object you have. In this paticular case, you can see the data for one stream line You may see the single stream line with graphics command:

 To find all such lines, you can use Manipulate to sweep over all possibilities: 
Eventually, you may use the following subroutine to extract the actual data points : Basically it first replaces the Arrowheads objects with Null and Arrow objects with List. It deletes the Null's and converts the list to a proper form with the Flatten function.  

The final result 

@Vitaliy, I am thinking about making a rug that looks like the picture in your post ! :-)

The easiest way is to add many random points in the range of plot. Remember the definition of the stream lines that they do not touch other, so you can just scatter points in the plotting field to make them the points that must be passed by a stream line. An improved vector field plot is like this:
The problem with this method is the low efficiency, the more points in a closed space, the more likely some of them falls on the same stream line. On the other hand, this also tells you that it is very computationally expensive to create a global dense stream plot. If you run the following code:you will find that Mathematica actually prevents itself from plotting a extremely compact field. If not, the process might kill the front end kernel.