Faces 4 and 5 definitely block the spell, and face 2 might also block it
Faces 4 and 5 both block spell effects, which clearly prevents Ice Storm from having any effect inside the barrier (when cast from outside it). Face 2 is more ambiguous: it says that
Nonliving matter can't pass through the barrier.
This should prevent hailstones from passing through the barrier, but the question to consider is whether the hailstones actually need to pass through the barrier. The spell's description says:
A hail of rock-hard ice pounds to the ground in a 20- foot-radius, 40-foot-high cylinder centered on a point within range.
The spell doesn't say where the hailstones come from, it just says where they end up: on the ground. One reasonable ruling is that they start at the top of the cylinder and fall straight down, landing on any ground below them. With this ruling, face 2 of the cube would protect its area from the spell (although the DM could rule that if the cube is switched to a face that doesn't block the ice, the ice falls down at that point and affects anything inside). Another equally reasonable ruling is that ice appears in all of the empty space throughout the spell's area and then falls down from there. In this case, face 2 will not protect you, since it doesn't block the spell that causes the ice to appear inside it.
In short, for face 2, you're going to have to ask your DM. If you are the DM, you're going to have to make a ruling.
Additional note: Moving the cube along ice-covered ground might cause something interesting to happen
The Cube of Force also states that:
If your movement causes the barrier to come into contact with a solid object that can't pass through the cube, you can't move any closer to that object as long as the barrier remains.
This means that if you have activated one of the faces that blocks the ice and you try to move, the barrier is going to be pushing up against the hailstones lying on the ground. I don't know if I would consider a layer of hailstones to be a "solid object", so once again DM needs to make a ruling on what happens in this case. They might rule that the ice effectively locks the barrier in place, or they might rule that it counts as part of the ground, which you can allow to pass through the barrier unimpeded. They might even allow the barrier to act like a snowplow, perhaps with a strength check required to push the ice out of the way.