When is impedance matching necessary? Isn't it much less efficient on all other aspects?
It took me a while, but I think I finally understand the nature of your question.

You seem not to understand that you cannot measure a transmission line's Z0 with an Ohm meter.

A transmission line's impedance is determined by the ratio of its electric field to its magnetic field. This is determined by the line's physical dimensions, not by the materials used to build the line.

A coaxial cable's characteristic impedance, for example, is determined by the ratio of its conductor diameters and we ignore the resistance of the conductors use to build the cable (within reason). A short piece of 50 Ohm cable will typically have conductor resistance values in the micro Ohm range.

We use impedance matching in circuits when we need to improve the power transfer between 2 points in the circuit. You asked "when does impedance matching become necessary", and the answer to that depends entirely on the situation. It may be the case that a high power circuit will burn out if the magnitude of the reflection coefficient is greater than 0.2, but this amount of reflection can usually be tolerated low power circuits. 

In response to the questions below: To research transmission line impedance, search on phrases such as microstrip, stripline, or microstrip or stripline calculator.

Here is a Wikipedia article.  http://en.wikipedia.org/wiki/Microstrip 

A simple example would be if you were to drive a 2 Ohm load with a 50 Ohm source. Without impedance matching, only 15% of the power would be deliverd to the load. 

You can match this load to the source with a 1/4 wave, 10 Ohm transmission line. This match will be perfect at the frequency where the transmission line is 1/4 wavelength, so 100% of the power will be delivered to the load at this particular frequency. At other frequencies, the match will be degraded.

Response to 2nd question:
You made 2 mistakes. First, in the calculation you made, the voltage is only 4%, but the power is proportional to V^2. But this is irrelavent because you can't calculate the power transfer this way.

Think of it this way. The impedance of a free space is 377 Ohms. If we connect an antenna to this 377 Ohm source, we don't treat the 377 Ohms as a dissipative loss point, but rather an impedance that dictating the ratio of the E and H fields, nothing more.

The correct way the calculate power transfer is to calculate Rho, the reflection coefficient. Rho = (Z0 - ZL)/(Z0 + ZL). For my example Rho = (50 - 2)/(50 + 2) = -0.923 Power transfer is 1 - Rho^2 = 14.8%