Is there any problem with the idea that gravity is repulsive at long distances? If gravity is repulsive at long distances, would this suffice in explaining "dark energy"? Was there any reason why a new concept (dark energy) was introduced rather than just assuming that gravity becomes repulsive over long distances?
 A: Gravity, per general relativity (GR), is normally attractive. Normally means that the sources of the gravity, and thus the sources that determine the geometry and curvature of spacetime, have positive energy density, and obey other positive energy conditions. The pressure and other factors that enter into the stress energy tensor that is the source of the curvature of spacetime can, however, be negative for so called exotic matter, or exotic fields. In those cases those factors contribute to a repulsive effect in gravity.
But it is not repulsive then when far away, it is then repulsive everywhere. The negative pressure contributes everywhere, not just at a distance. 
That is a somewhat simplistic description of what happens. Let's try to be more specific. To be a little more accurate, GR is described by an equation that reads, in non-mathematical terms,
Einstein Tensor =  (Cosmological constant term) X Curvature Scalar X metric tensor + constant X Stress Energy tensor
The left hand side represents the curvature while the right hand side is the stress energy tensor representing matter and energy, and the cosmological term. The cosmological term is said to represent the dark energy. 
Nobody know what the dark energy is, it's a constant value of dark energy per unit volume, and it is very very small. It seems to be constant, measurements indicate no variation on what is observed further away than closer in (i.e., from radiation emitted a long time ago or more recent). The hypothesis is that it is some exotic quantum field, similar to the field that caused the inflation soon after the Big Bang, but different, with negative pressure, causing the accelerated expansion of the universe. Some theorize that it might vary some over space and time (theory called quintessence, appropriately), but none of that has been observed. Fact is that we don't know, but if it is due to a quantum field it would be an exotic one. Others theorize that GR needs to be modified, but none of the modifications proposed have found any evidence in their favor, so far.
It is worth noting that the cosmological term is so small that it has in essence no effect on gravity's effect for anything except cosmological distances. Black Holes form everywhere in the universe, and that little repulsive effect makes no difference. 
The reason it makes a difference cosmologically is that if the term is a constant energy and repulsive pressure per unit volume, as it seems to be from observations, as the universe and thus the volume expands, there is more of that energy and negative pressure. When it becomes large enough it becomes more of an effect than that of positive energy matter, pressure, and energy. That is what's happened in the universe, where now dark energy contributes 68% of the total energy content of the universe, and so dominates. As a result we are accelerating in the expansion. But it is NOT an effect ONLY a long distance away, it is everywhere in spacetime. 10-12 billion years ago, and until about 5 billion years ago, matter dominated, everywhere. Farther back in time radiation energy dominated, again everywhere. 
A: There are a few problems I can think of with this idea - 


*

*Gravity has to, at some distance turn from positive, to zero, to negative. It would be interesting to know that distance.

*Will the repulsion increase, or decrease with increasing distance? Dark energy hypothesis indicates repulsion would go up with increasing distance. Which does not make sense - what would make the repulsion increase on the way? If something (say space) causes it to increase, then it is not gravity in true sense, because gravity is caused by mass.

*Gravity is understood as curvature of space(time), caused by mass. How mass is expected to cause a reverse curvature at larger distances? that even, increasing with distance. Meaning no limit, the repulsion can be infinite at infinite distances. We can be positive that there is some mass at infinite distances from every mass in the universe. That means every mass should be repulsed with infinitely large strength.
