You are mistaken. Economics doesn't actually concern itself with measurable quantities, but rather with the behavioral functions that give rise to measurable quantities (such as price, investment, etc). The most obvious example of this is the utility function, as they are unique up to scale – meaning that there are several (actually infinitely many) which nonetheless do the same thing.
Consider the prisoner's dilemma, for instance: we might introduce for each player a scale factor which multiplies his whole payoff matrix. But no matter how we change those payoffs, so long as the structure of those matrices is preserved so will be the behavior of the prisoners. So nothing really changes. But that is fine, because what we're not interested in modelling is the value of the choices to each player, but how the players choose.
So it isn't necessary to introduce any concept of the measurable economic value of anything so long as we might say that it is preferable to something else. Which is to say that we might model the behavior of any agent without introducing a cardinal measure: we can do fine if his choices are merely ordered (this is the idea behind Revealed Preference, which was initially formulated by Paul Samuelson as an alternative to utility functions, but which was later on found to be equivalent to them).
But to answer what's bugging your mind: there isn't practically anything even mildly relevant to human life which economists won't deal with. Because in so far as people's choices are observable, they can be treated by an economic model. For instance, there's a chapter in the Handbook on the Economics of Leisure which is called The Economics of Sleep and Boredom. I've also seen a paper once on the economics of suicide (there's actually a small literature on this). Another example – though less odd and more important – is Becker's theory of crime, which takes crime out of the domain of other social sciences (which might explain it in terms of marginalization, poor education, etc) and into the domain of expected utility and choice theory (i.e. what are the odds the guy think he has of getting away given the circumstances).
Contrary to popular opinion, dealing with money and crunching numbers is often the least important and most boring part of an economist's job. Most often economists are concerned with figuring out why people would do what they do.