What is the meaning or purpose of inertial motions It seems to me that, except for light, inertial motion is never possible in real world where there is gravity everywhere. It is at best some kind of idealization.
But why inertial motions are so prominent as part of Special Relativity and the Lorentz transformations etc? 
It seems that the whole notion of Lorentz transformation hinges on having inertial frame while in reality there can be no such thing as inertial frame given the presence of gravity. 
 A: For any isolated object, i.e. one that is having an external force applied to it, the frame that object lives in is locally inertial. By locally I mean that if you consider a small enough region of spacetime around the object it will be impossible to tell the frame isn't inertial. How small this volume is depends on how curved spacetime is. This is an exceedingly important principle in general relativity or indeed in anything that uses differentiable manifolds.
Another way of putting this is that even in general relativity, if you consider a small enough region the physics can be described to arbitrary accuracy by special relativity.
So while you are correct that there is no such thing as an inertial frame (unless you're lucky enough to be doing physics in a universe free of matter) the concept is a very useful one. In many cases frames are close enough to inertial that we can ignore the deviations. After all, quantum field theory uses only special relativity but it can make the most accurate predictions ever made by a scientific theory.
Response to comment:
I'm not sure what you mean by The motion of inertial frame is observable. An inertial frame is not an object, it's a choice of coordinates. Motion relative to an inertial frame is observable.
We know our frame is inertial if all the objects moving relative to it obey Newton's first law. That is, if we set something moving and apply no force to it then it will continue moving in a straight line at a constant speed.
Let's consider a specific example. Suppose I'm floating around in the ISS, and I place a ball at rest relative to me but a metre nearer the Earth than I am. The ball will just stay there, so I could be forgiven for thinking that the coordinate system I'm using to locate points inside the ISS constitutes an inertial frame. And for all practical purposes it does.
The trouble is that the ball is actually in a slightly (one metre) lower orbit than I am and its orbital period is therefore shorter. If I watch the ball for a long time I'll see it begin to move relative to me i.e. it isn't obeying Newton's first law and I can tell my frame isn't inertial. But the period of the ball differs from mine by only about 1 part in 20 million so I'd need to watch for a long, long time.
So the point is that to within a very small error the coordinates I've set up inside the ISS do constitute an inertial frame within a radius of a metre from the origin. But suppose I was doing some very sensitive experiment where that 1 in 20 million deviation was significant. Well I could make the experiment one millimetre in size instead of one metre, and that would reduce the error to less than one part in a billion.
And so on. I can make my frame approximately inertial to whatever accuracy I want by reducing the size over which I use it.
