How do you solve a net ionic equation without a balanced equation?
This sort of question is really all about solubility. What you're really asking when you're trying to get a net ionic equation is What combinations of the ions are insoluble? Net ionic equations involve only ions that end up as precipitate on the right side of the equation. If you do enough of these problems, you get a feel for the general solubility trends of common ions, but there should be a table in your textbook or have a look here. 

For this specific example, salts with alkali metal and ammonium cations are soluble. Nitrate and acetate anion salts are soluble. Therefore, all of those end up as spectator ions and we're left with just $\ce{Ba^2+}$ and $\ce{CrO_4^2-}$. Chromate salts other than those of alkali metal cations or ammonia are insoluble (rule 8 in the link), so the net ionic equation is:

$$\ce{Ba^2+_{(aq)} + CrO_4^{2-}_{(aq)} -&gt; BaCrO_{4(s)}}$$

In this case, it's already balanced, but that may not necessarily be true.