Yes, there does. If you do this in the context of fuzzy set theory (one possible way to extend things), there exist two different notions of cardinality. There's an entire book on cardinality in fuzzy set theory actually. Scalar cardinality comes as the relevant concept here.
For a possible example of such a set consider the set of your eight closest friends in the set of "tall people" (or your "Top Eight" in your MySpace page if you had one a few years ago). I'd at least guess that, you'll have fractional "degrees of membership" for each person in the set of "tall people" as say a 6 foot person, I would expect, you would consider only somewhat tall and thus perhaps classify as .8 tall or something like that. E. G. say I have friends Bob, Sally, Sue, Steve, Chris, John, Elmer, and Guido, with degrees of membership in the set of tall people of .3, .5, .6, .2, .86, .35, .6, and .1 respectively. Then, then scalar cardinality of the set of my friends in the set of tall people equals 3.51 out a possible 8.0.
As another example, go to the room where you have the most books in your house. Find the most common book cover color where each book gets assigned one of {red, blue, green, cyan, magenta, yellow}. Now consider each book according to how red/blue/green/cyan/magenta/yellow that book is. So, one book cover might have a degree of membership of .6 in the set of red books, and another .1 degree of membership. Finally, sum all these together, and at least in plenty of cases the set of say red book covers in that room will have a non-natural number for its cardinality.
In my opinion, if you think about things in terms of "how well does x fit with y", it doesn't come as all that hard to find examples of real-world sets with fractional scalar cardinalities (or more generally it's NOT hard to find fuzzy sets). In my opinion, it actually comes as harder to find real-world sets with natural number cardinalities, but I've almost surely started to digress.