Because you have to match up the + and - irrationals I don't think you get a longer smallest stick doing this; although obviously you can have the same-length smallest stick.

I've written a thing that looks for integer solutions; but it's slow and crap and obviously "I searched exhaustively and found this" is much less interesting than "I thought really hard about it and realised that all solutions look like this". Also it can't handle irrationals and so far can do fractions only where all stick-fragments share a denominator....