Length contraction in special relativity: can space be at rest in any frame? Suppose a rod is moving at speed $v$ relative to me along its length.
$L_0 = {}$length of the rod in the frame in which the rod is at rest
$L = {}$length of the rod in my frame
Then $$L = L_0 \sqrt{1-\frac{v^2}{c^2}} $$
Let us now consider another scenario. I am moving towards a star at speed $v$. Then the distance between me and the star is given by $$L = L_0 \sqrt{1-\frac{v^2}{c^2}} $$
My question is:
What is the meaning of $L_0$? I understand the meaning of a rod being at rest in some frame. But what is the meaning of distance between me and the star being at rest in any frame? Can space between two points be at rest in any frame?
 A: There is no frame in which "space" is at rest.
Using the length contraction formula
$$
L = L_0 \sqrt{1 - \frac{v^2}{c^2}}
$$
for the distance to the star is slightly misleading. The distance $L$ in your frame corresponds to two points, $x_0 = 0$ for you and $x_1 = L$ for the star, that you are observing simultaneously, say at time $t_0$. In the star's frame the same locations will have coordinates
$$
x'_0 = - \frac{vt_0}{\sqrt{1 - \frac{v^2}{c^2}}}\\
x'_1 = \frac{L - vt_0}{\sqrt{1 - \frac{v^2}{c^2}}}
$$
and you'll be tempted to say that the distance in the star's frame is 
$$
L_0 = x'_1 - x'_0 = \frac{L}{\sqrt{1 - \frac{v^2}{c^2}}}
$$
or that 
$$
L = L_0 \sqrt{1 - \frac{v^2}{c^2}}
$$
But in the star's frame, locations $x'_0$ and $x'_1$ are actually observed at different times, reading
$$
t'_0 = \frac{t_0}{\sqrt{1 - \frac{v^2}{c^2}}}\\
t'_1 = \frac{t_0 - v x_1 /c^2}{\sqrt{1 - \frac{v^2}{c^2}}} < t'_0
$$
So this $L_0$ is not a distance in the proper sense of the word. 
The important thing to learn here is that this is a result of relativity of simultaneity: events/locations/measurements that are simultaneous in one frame will not be simultaneous in another.
In fact the same thing happens for the rod, but at least in that case we do have a meaningful "rest length" in the rod's frame. 
A: First:  You've chosen to express "the distance from me to the star" as some formula involving a quantity called $L_0$ --- and then you've asked us to tell you what $L_0$ means.  But you're the person who wrote down this expression, so only you can know what you meant by it.
Second:  It's very hard to tell what you're actually asking, but I am pretty sure that I can identify the source of your fundamental confusion.  Namely:  The length of the rod (as defined in your frame, the rod's frame, or any other frame) does not change over time.  The distance from you to the star does change over time.  You seem to be trying to treat the length of the rod and the distance to the star as perfectly analogous, but they are not analogous at all.  
Third:  If two observers are both present at an event $E$, and if you know the coordinates that the first observer assigns to an event $F$, you can Lorentz-transform those coordinates to find the coordinates that the second observer assigns to event $F$.  You seem to be imagining two observers, one at an event $E$ on the spaceship and one at a different event $E'$ on the star, and trying to Lorentz transform one set of coordinates to the other by blindly applying a formula.  
So start over:  You are on the ship.  Your clock strikes 1:00.  Call this event $A$.  At that moment (according to you),  a clock on the star strikes 1:00.  Call this event $X$.  At that moment, according to an observer on the star, your clock is striking 2:00.   Call this event $B$.   
"The distance from you to the star" depends on both the observer and the time when the observation is made.  To you, at 1:00 by your clock, "the distance between you and the star" means the distance, in your coordinates, from $A$ to $X$.  To you, at 2:00 by your clock, "the distance between you and the star" means something else.  To your friend on the star, at 1:00 by his clock, "the distance between you and the star" means the distance, in his coordinates, from $X$ to $B$.  Before you can start comparing one of these distances to another, you have to decide which two you're trying to compare.  
Finally, with regard to the time it takes light to get from the star to you --- remember that the light currently arriving at your ship left the star at a time when the star was further away than it is now.  Light currently leaving the star will arrive at your ship at a time when the star is closer than it is now.  So if you want to talk about "the time it takes light to get from the star to your ship", you need to decide not just who's measuring, but which light beam's travel times you're talking about.  (Or perhaps you mean some other travel time.)  
Bottom line:  Different questions have different answers.  Questions like "How far does the guy on the star say he is from me?" have many possible meanings, each of which yields a different answer.  The key thing you are missing is that you are not being clear (certainly not with us, and I suspect not with yourself) about what you're trying to ask.
A: Distance contraction is a case of length contraction. The length of a rod in its rest frame is contracted for observers moving near light speed. Distance contraction is working in an analog way: The length of the rod corresponds to the distance between a mass object at the point of departure and a mass object at an end point. 
For this purpose we must suppose that both points are belonging to the same frame, that means that the relative velocity between the two mass objects at the starting point and at the end point is zero (or at least non-relativistic). For practical purposes, this definition is in most cases sufficient, for example if you want to define a distance between Earth and an exoplanet. An astronaut moving from Earth to the exoplanet will have to take into account the distance contraction. As a result, he may reach exoplanets within his lifetime which he could not have reached without the phenomenon of distance contraction.
In contrast, the astronaut traveling near light speed and the exoplanet moving at non-relativistic velocity with respect to Earth cannot be considered as one and the same reference frame because their relative velocity is relativistic. That means that your example does not work: if you are approaching a star at relativistic relative velocity, you are not in the same reference frame as the star. No rod can be held between you and the star.
