How can the surface of a liquid exist if there are no attracting molecules above it? Okay this is probably a stupid question but here goes: 
We know that molecules in a liquid are attracted from all the molecules around it so there is no net attraction. Well, then how do the molecules at the surface exist because there are only molecules below it and not above it? Won't they accelerate downwards? 
Also, does this have something to do with surface tension? If it does, then isn't surface tension a force that acts along the surface and not into or out of it?
 A: They do not accelerate downwards because the attracting force is balanced by the repulsing force. At larger distances the molecules attract, at shorter they repulse. The repulsion force is due to Pauli exclusion principle.
A: The short answer is that there are molecules above the surface of a liquid. Its the atmosphere! There is atmospheric pressure, which is the countless number of collisions that Nitrogen, Oxygen, and other molecules the make up the air that collide with the liquid, pushing down on it. If you don't have an atmosphere, for example in outer space, the liquid will immediately leave the container because there's no force keeping the molecules in the container. Where does this downward force come from? See next paragraph.
There's also gravity. This makes everything accelerate downwards, where down is the direction pointing from the surface of the Earth to the center of mass of the Earth, which is somewhere near the center of the Earth.
It does have something to do with surface tension. When we think too exact, we come to the issue of thinking that if there's only a horizontal force, then there's no way there's a vertical force! This is the idealization issue your experiencing. Really, there is a component which acts vertically, because the surface of a liquid is not exactly flat.
A: When the molecules get close enough to each other, the net force between them becomes repulsive.  You can think of this simply as the molecules colliding with each other — insofar as it's meaningful to speak of collisions between such things as molecules, that's basically what's happening.
A: This may or may not directly relate to the topic but I think it does.I have noticed when a cup of coffee is poured(esp with cream) and stirred the bubbles are large or small.The ones swirling in the centre disappear first(?pulled or pushed by surface tension). My observation is if that if the barometric pressure is high or rising(good weather) the bubbles are larger in the centre and periphery.(and stay in the centre longer) versus low pressure where the bubbles are smaller and disappear from the centre more quickly as do the smaller bubbles at the periphery.In other words does the barometric pressure either raise or lower the surface tension and subsequently the size of the bubbles and their position on the fluid surface. This may be totally wrong but they say "there is no stupid questions"...Im not sure I agree!
A: 
how do the molecules at the surface exist because there are only molecules below it and not above it? Won't they accelerate downwards?

You could ask the same question for the surface of the earth.Why is it not imploding? The answer is "quantum mechanics". When one is talking of molecules and atoms one has to talk in terms of quantum mechanics. 
Atoms exist because the electrons orbit around them in very specific quantized orbits which might be distorted when new forces enter but need a lot of energy to be destroyed, which gravity does not supply, as a very weak force.
 All interactions in liquids and solids and gases between atoms and molecules are electromagnetic even though the atoms and molecules are neutral, because there are spill over electromagnetic fields from the shape of the electron orbits of the atoms/molecules. The spill over fields may be attractive and create the coherence of molecules  and explain the creation of solids.  In liquids these spill over attractive forces create surface tension, and the horizontal (actually surface of sphere with radius to the earth center) level of the liquid.
When external forces become large, the electron orbits come to play with their negative electric fields which repulse each other; also the Pauli exclusion principle assures that only one electron can be in an orbit, so compressing find resisting forces .
For a liquid surface the compression comes from the force of gravity, as @QEntanglement explains, the weight of the column of air over the liquid.
