Thank you for these helpful comments!

I don't think finite unions is that easy. When you are partitioning the set $A \cup B$ into pieces you must be using half-open intervals. That means that elements of your partition may have a combination of points from $A$ and points from $B$. You can't get all of the points in $A$ to go in $[0, \frac{\epsilon}{2})$ and all of the points of $B$ to go in $[\frac{\epsilon}{2}, \epsilon)$. Maybe the way you can get the points of $A$ to fit in a small interval is incompatible (somehow) with the way you can get the points of $B$ to fit in a small interval.

I don't know whether the union of two small sets must be small, but suspect that it must. (That there is no Banach-Tarski decomposition in one dimension, using only finitely many pieces, seems like it might be relevant here.) Since countably many translates of a Vitali set covers the line, this notion of small is not closed under countable unions.