Unclear this is best, but: $r=\sqrt{3}/2$.

Unclear this is best, but: $r=\sqrt{3}/2$.

JukkaKohonen's suggestion:

I made no attempt to optimize, but this certainly shows the smallest sphere has radius strictly less than $\sqrt{3}/2$.

Seems unlikely this is best, but: $r=\sqrt{3}/2$.

Unclear this is best, but: $r=\sqrt{3}/2$.