Why does Nature use a 4-level system to encode information in DNA?
The current hypothesis is that RNA came first, DNA and proteins came later. So the reason that four bases are used might be related to the initial RNA world, and then DNA just reused the already existing RNA bases in a slightly modified form. In the RNA world, all functions had to be performed by RNA. Having more bases available than two would likely be important to be able to adopt various structures and create binding pockets or active sites for ribozymes. 

You can't really think of the genetic code as an abstract data storage device. There are physical and chemical consequences to the choice of encoding. For example, proteins have to be able to bind to DNA and recognize particular patterns. With your binary code, the recognition sequence would have to be longer, because each basepair contains less information. The tRNA anticodons would have to be larger for protein biosynthesis to work with the binary code. Another issue that plays a role in some processes is that GC base pairs are more stable than AU/AT base pairs.  

These are all just hypotheses. Evolution doesn't necessarily choose the best option, sometimes it is just the most convenient one that still works well.

I also found a review titled "Why are there four letters in the genetic alphabet?" that makes a similar point as my first one. 


  All present models to explain the fact that we have four base types in
  our genetic alphabet hinge, in covert or overt form, on the assumption
  that the genetic alphabet evolved in an RNA world


Another factor I didn't think of that is mentioned there is that while more bases make better ribozymes, more bases also decrease the accuracy of  replication. 


  In summary, two-dimensional RNA-like structures (and, presumably also
  the three-dimensional structures) become better defined as alphabet
  size increases, whereas the accuracy of replication decreases.