The answer is $\frac{\sqrt 5}4$. The idea is to center them between two vertical lattice lines and move vertically. If you then draw circles with that radius around all lattice points, the remaining area is essentially a bunch of stars inside lattice squares. It's clear to see that you can only achieve a distance of a half by being on the edges so increasing the radius isn't possible.

gvole wrote: The idea is to center them between two vertical lattice lines and move vertically. Could you elaborate this part a bit more? And how do you obtain the upper bound?