Is there any reason an arbitrary DNA sequence such as 5'-ACACACACAC-3' would not exist in nature?
You must first understand a few things:


What are proteins made of
What is DNA made of
What is the relationship between DNA and proteins


From DNA to proteins

DNA is transcribed into RNA which is then translated into protein. The protein, then has some effect (e.g. could be an enzymatic effect) such as affecting the level of sugar in the cell, digesting some lipids or affecting the expression of other genes.

More information on Khan Academy > Central Dogma

What is DNA made of?

DNA (or at least the genetic code) is made 4 different molecules which are called nucleotides. We abbreviate their names by the letters A,T,C and G.

What are proteins made of?

Proteins are made of amino-acids. There are ~20 different amino acids.

Genetic code

For the interest of your question it is now essential to understand how does the succession of nucleotides A,T,C,G in the DNA is reflected in the protein. This is called the genetic code.

We call a series of three nucleotides, a codon. AAT is a codon for example. Each codon (except special codons) correspond to a specific amino acid. This correspondance is called the genetic code. Here is the universal genetic code (where the T has been replaced by U because RNA contains U instead of T)



Consider for example the codon UAU and AUU (where only the order changes) The first one codes for the amino acid Tyr while the other codes for Ile. So yes, the order matters.

Information theory

Globally speaking, for any information the order of a sequence is of fundamental importance! More information on Khan Academy > Information Theory.