How sterile is sterile when working with nucleic acids to prevent contamination?
Short answer: Yes

Long answer: Depends what you are working with. 

DNA: If you are working with DNA, its pretty stable and you can usually get away with a 70% ethanol wash/autoclave (mainly to prevent contamination and obtain consistent results). EDIT: Read Chris's answer also below

RNA: If you are working with RNA well.. whatever you did for DNA doesnt apply anymore. You have to take it to the next level.  You will need to replace everything, you'll need nuclease free water, tubes, reagents and whatnots. Throw out ethanol and bring in RNAZap, DEPC water or something like that.


  An RNase free environment is essential when working with RNA samples. There are two main reasons for RNA degradation during RNA analysis. 
  
  First, RNA by its very structure is inherently weaker than DNA. RNA is made up of ribose units, which have a highly reactive hydroxyl group on C2 that takes part in RNA-mediated enzymatic events. This makes RNA more chemically labile than DNA. RNA is also more prone to heat degradation than DNA. 
  
  Secondly, enzymes that degrade RNA, ribonucleases (RNases) are so ubiquitous and hardy; removing them often proves to be nearly impossible. For example, autoclaving a solution containing bacteria will destroy the bacterial cells but not the RNases released from the cells. Furthermore, even trace amounts of RNases are able to degrade RNA. Note that RNAses are present everywhere (skin, reagents, normal plastic etc).
  
  Therefore, it is essential to avoid inadvertently introducing RNases into the RNA sample during or after the isolation procedure.


Remember that following microbiological aseptical techniques is usually good practice (and a requirement) when working in a molecular biology lab.

There are excellent guides available online for working with RNA:

I took the above paragraph from here: 
http://www.bioline.com/us/rna-hints-and-tips

Life tech has a few more tips for you: https://www.lifetechnologies.com/uk/en/home/references/ambion-tech-support/nuclease-enzymes/general-articles/working-with-rna.html

Here is a good pdf to have in your labbook as a reference sheet:http://genomics.no/oslo/uploads/PDFs/workingwithrna.pdf

If this scares you , that's good because when working with RNA there are a million things that can go wrong the first few times you try it. You have to be very careful, and adapting a cavalier attitude such as not bothering is going to come back and haunt you at night...