Why doesn't national savings, in aggregate, always equal to zero in monetary terms? Referring to this question and this one
Imagine you had a closed economy with two people and they both started off with $100 dollars. How would net savings in aggregate ever not be zero? Anything spent, net, by person 1 would be a positive savings for person 2, and a negative savings for person 1. The same is true vice versa.
I understand that if person 1 and person 2 both produced 10 bushels of wheat and then ate only 5 of them, and then ground up the other five and used them as fertilizer (bad example, I know) that they would have 'saved' and 'invested' 10 bushels of wheat in aggregate. However, in monetary terms, the personal savings rate in aggregate would still be zero because they didn't sell the wheat to anyone and thus had no personal income (again, in dollar/monetary terms).
So, to sum up, my first question is, in monetary terms, in a closed economy will aggregate savings always be zero?
Second, how does the BEA get a positive savings rate, and how do we have positive gross national savings, in addition to running a trade deficit?
I think this boils down to misunderstanding what saving in macroeconomics is. Saving does not cancel each other out. In closed economy, private saving is difference between income (which is by definition equivalent to output so I will be using output and income interchangeably) and consumption $S=Y-C$ and public saving is difference between government spending and taxes $T-G$. In your case there are only private individuals so let us assume there is no government so that $T=G=0$ and consequently national identity will be given just by $Y=C+I$ (also note this shows that investment and saving must be equivalent since $Y=C+I \implies Y-C=I \implies S=I$). See Blanchard et al Macroeconomics a European Perspective for further explanation of those identities.

You you can't just save without having income. The money you saved in previous time periods would be part of your net wealth. For example, if in 2010 nothing is produced and sold both individuals have no income  and thus cannot form any new saving. In this example saving for 2010 would be 0. The money they hold from previous time period would be their net wealth.   However, note in this case there cannot be any consumption either and consumption will be zero. Why? If there is no income and production there cannot be any consumption as well. If something would be produced in 2009 and it was not consumed in 2009 so that it is still avaiable for purchase in 2010 it would be recorded as an inventory investment. So in your example, no matter what those people do $Y, C$ and $I$ are all zero $Y=C=I=0$.

If they actually produce something saving is totally possible. Consider the following trivial  example, suppose the person A produces in 2010 $\\\$75$ worth of widgets and person B produces in 2010 $\\\$50$ worth of food. This means that the national income in 2010 will be $Y=50+75={\\\$}125$. Now let us suppose person A buys all food from person B for $\\\$50$ and person B buys $\\\$50$ of widgets from person A. In this case consumption for B is $C={\\\$}50$ B has zero savings since for $B$ $Y_B-C_B=0$. However, A will have some saving since for A income was $Y_A={\\\$}75$ and consumption was $C_A={\\\$}50$ so there is $S_A=Y_A-C_A={\\\$}75-{\\\$}50={\\\$}25$ which on national accounts will turn up as and inventory investment. So everything balances since $Y=C+I$ holds as we have $Y_A+Y_B ={\\\$}125$

Note it makes no difference whether everything is expressed in purely monetary units or real terms. In above I assumed all those quantities are expressed in nominal, so monetary terms, but I could as well just make them real by deflating those prices by some deflator. The equality would still hold equally well.