Are length contraction and time dilation physical? Are length contraction and time dilation real physical phenomena, or are they just a measurement issue? 
 A: It's unclear what you mean by "measurement issue" here, but my reading suggests that your question is loaded with what most modern physicists would consider a false dichotomy, namely you implicitly differentiate between "real physical phenomena" and "measurement issues."
A modern physicist would view a phenomenon to be "real" precisely when it has measurable consequences.  As a result, we might as well take "a phenomenon having measurable consequences" as the definition of a "real" phenomenon.
Time dilation and length contraction both have measurable consequences, so are they are real by this definition.
For example, if we insist on defining time operationally (namely using clocks which are real physical objects that exhibit repeating behavior), then we find that time dilation has measurable consequences.  See, as a particularly clear example, this experiement.
A: As mentioned by the popular physicist Brian Greene, all objects are constantly moving within that 4 dimensional structure known as Space-Time. The speed of that constant motion is the speed of light, be it motion across space, across time, or across Space-Time in any other possible direction.
If you analyze the outcome of such a phenomena, you end up with Special Relativity and all of its equations. 
It also makes it crystal clear as to the relationship between "Actual" and "Measure". If you have some time available, then Go to http://goo.gl/fz4R0I and watch the "KSP Special Relativity" video collection. Its a slow step by step analysis of motion, but it leads you to all the equations, including therefore the Time Dilation equation and the Length Contraction equation, while at the same time reveals the absolute foundation that creates the relativistic outcome. ( Total time of 9 short videos = 1 hour 39 minutes )
If you look at a 2D picture of a thin horizontal wire, you see it's length. But if the wire is then rotated 45 degrees on the Z axis and a new picture is taken, in the new 2D picture of the wire, the wire now seems to be shorter in length. All of this is due to a percentage of the wire now also extending across the Z axis.
In short, objects exist within the 4 dimensional environment known as Space-Time. These objects can be rotated within this 4 dimensional environment. However, we are confined to viewing only 3 Dimensions, and we do so since we can not look across the dimension of time. We are confined to the present time only. Thus when an object rotates within Space-Time it appears to contract in spacial length even though its actual length has not changed at all, just as the length of the wire has not changed at all.
A: Length contraction and time dilation is due to one viewing a moving thing from a different frame of reference.  A space-ship that is 1000 feet long, will to its occupants, or even a parallel traveller, remain 1000 feet, but stationary observers might see it contracted at 900 feet.  
