How close do you need to be to a Poké-Stop?

You need to have the PokéStop or Gym inside your "action circle" -- the pulsing circle that will sometimes spawn. This circle's radius at its widest point is 40 meters, so you must be within 40 meters of the building you want to interact with. This radius is set to the same as Ingress'.

When a building is eligible, you'll notice that Pokéstops will "expand" out, and the action circle will leave a trace mark on the ground.

When walking you see a purple circle around your avatar, if the pokestop is inside that circle you can activate it.