If your notion of constructive considers the family of binary sequences as uncountable, then set $a_0=1$ and consider the family of all sets $\{1,a_1,a_2,\cdots\}$ with the property that $a_{i+1}$ is either $2a_i$ or $2a_i+1.$ If you do not allow "lawless" binary sequences then there are not likely to be any uncountable families. 

This is essentially the binary version of Valerio Capraro's answer, but it looks like it has less baggage.

Another point of view: label the root of an infinite binary tree 1 next level 2 3 the four nodes below that 4 5   6 7 (left to right) and so on. Each set corresponds to a unique path starting at the root.