Using the information from http://en.wikipedia.org/wiki/Parallel_(geometry)#Extension_to_non-Euclidean_geometry:

2. parallel, if they do not intersect in the plane, but have a common limit point at infinity, or [...]

So, if we put this in a non-Euclidean plane, it can be done with one line:

But, of course, this would take forever to draw (But it does pass through the centres of all stars)

Using the information from http://en.wikipedia.org/wiki/Parallel_(geometry)#Extension_to_non-Euclidean_geometry:

2. parallel, if they do not intersect in the plane, but have a common limit point at infinity, or [...]

So, if we put this in a non-Euclidean plane, it can be done with one line:

But, of course, this would take forever to draw (But it does pass through the centres of all stars)

I can also confirm it is impossible to do on a Euclidean plane without going outside the box with this Python script.

There are 84 different ways to do it with 5 lines inside the box, but that should be around 11 unique ones because most are reflected or rotated.