If Johann Schmitz measures the height of an oak tree in Bavaria in 1914 as 7.23 metres, and John Smith measures the height of a flagpole in Oregon in 2009 at 5.42 meters, we can know with confidence that the oak tree in Bavaria really was taller in 1914 than the flagpole in Oregon was in 2009*. We could even make these measurements in different nominal units across time and space: I know that 15.81 cubits is taller than 26.94x10-2 furlongs.
*Yes, I know about Special and General Relativity. But relativity doesn't complicate the relationship between observed units of measure and the underlying physical reality all that much. And even between observers of significantly varying velocity and/or acceleration, we can very precisely determine important characteristics of physical reality from their measurements in unqualified units.
Dollars, lira, deutchmarks. These economic units are also socially constructed abstractions. There are no dollars in nature. But, more importantly, a dollar by itself does not even correspond to anything real. If Juan Ferrari measures the Gross National Product of Italy in 1922 at 7.32x1012 lire, and Jean Lefèvre measures the privately held debt in France in 1989 at 3.768 francs, I know absolutely nothing. We have no way of knowing even whether or not the underlying reality differs by several orders of magnitude either way! Even if we measure the same quantity in the same nominal units at different times (e.g. the GDP of the United States in dollars in 1932 and 2009) or the same quantity at the same time with different nominal units (e.g. GDP of Germany and France in 1945) we still can't directly compare the quantities. A dollar is different in 1932 than 2009.
Not only are nominal economic units different, they are different in ambiguously defined ways. It's not that we absolutely cannot compare dollars in time or different currencies in space, it's that there are many different methods of making comparisons, and each method depends on a number of assumptions that are difficult to empirically justify. For example, when we compare dollars across time, economists usually adjust for inflation. But inflation itself is impossible to measure directly; we have to make a number of actual observations (the price of a loaf of bread, a pound of coal or oil) and apply fairly complicated — and controversial — models to compute the relative inflation.
There's nothing wrong with complicated models per se. However, I don't think any competent scientist would endorse using a complicated model to establish her primary units of measure.
Most economists are very smart, and I'm certainly not the first person to notice this issue. The problem is that there is no really good way to establish consistent economic units across time and space. Currency units at least afford precision and accuracy: I can go to the grocery store and precisely and accurately measure the price of a loaf of Hostess Wonder Bread™... even though I don't immediately know what that measurement means.