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 Caparo's answer, but it looks like it has less baggage.