if a vector has a magnitude equal to zero, can that thing exist? Can that thing be measured? consider a vector (a vector is a physical quantity ) having magnitude equal to zero.Now, if something has a magnitude equal to zero, can that thing exist? Can that thing be measured?
if net force on a particle is zero, then net force being a zero vector , will have some arbitrary direction ( according to definition of zero vector). is there any sense to say that the net force( in the above case) has some direction?
          If the magnitude of the electric field is zero in a certain region of space , then there will not be any effect of electric field on a charge moving through that region. Thus , if there are not any physical effects of a quantity ,is this right to say that the physical quantity exists? If your answer is no , then how can we say that zero vector exists?
 A: 
If something has a magnitude equal to zero, can that thing exist?

Consider the electric field.  It seems to me that if the magnitude of the electric field is zero, it is still the case that the electric field exists.  Otherwise, to what would the value of zero be assigned?
That is
$$|\vec{E}| = 0$$
is a statement about something that exists and not nothing.
A: Assuming that you are OK with the existence of zero as a real number in the first place, there is no added discomfort in admitting a zero vector. To deny it because its direction is ambiguous is to commit a grave mistake: much like how you have negative reals and some sums therefore must come to zero, like $3+2+(-5)$, some vectors like $(2,1),(-1,1),(-1,-2)$ sum to $(0,0)$. 
Just like how in removing zero from the reals, to save the closure property of addition, you would have to eliminate all of the negative numbers too, if you want to remove the zero vector from the space you need to only look in one quadrant of the plane or one octant of 3D space, the space $\mathbb R^n$ being forced to become the space $(\mathbb R^+)^n$, and to outlaw vector subtraction.
Outlawing vector subtraction is a mistake because it means you can no longer take derivatives of vector-valued quantities, so you can no longer define velocity given position, or acceleration given velocity. That is a huge cost to pay when the alternative is just to be happy with the idea that you cannot move east or west when you are standing at the North Pole.
A: One uses the method of approaching a limit, the "epsilon delta" mathematical method. The limit is approached incrementally and gets infinitessimaly close .There are various examples in the link.
As zero will just be approached then also the vector directions are sustained.
If the problem is started with the premise "equal to zero" then direction for vectors will be undefined , but the value zero is a legal real number which can be assigned to any field. 
