Why do liquids separate in space with no gravity? I've seen videos of people in space (on ISS) who squeeze a bottle or something and liquid comes out, it then separates into smaller balls.
Why is this surely it should stay pretty much together because on ISS there's no gravity from the Earth (or more precisely: it is cancelled by the centrifugal force), so the liquid should attract itself?
 A: Because liquids, water in particular have a high surface tension. 
For this reason blobs of water tend to become spherical. Now, given that it starts as an elongated stream (say because it's pushed out of a bottle), the stream breaks up in different pseudo-spherical bubbles;  if an astronaut were to pour water very, very slowly and carefully he could create a single spherical blob.
The concept is not dissimilar to drops, but without the gravity.

A: A cylindrical column of a liquid is intrinsically unstable due to surface tension. This is called the Plateau-Rayleigh instability. Check out this pdf for some pictures and equations. When you squeeze a liquid out of a bottle, it initially has a somewhat elongated shape, and the instability develops. 
A: Consider a spherical drop of liquid floating weightlessly. Now apply some mechanical energy unevenly to it. You have the mechanical energy of internal fluid flows competing against surface tension, which seeks to restore it to the lowest energy spherical state. But
surface tension if not all that strong, if the internal energies within the drop are too large,
they can cause it to break into two of more droplets. The real key to getting large drops, is to launch it with minimal internal (fliud) energy.
A: Well, I haven't seen your videos so I don't know what precise conditions were present but here's my guess.
It depends on how precisely they squeezed the bottle. It's true that if you had a ball of liquid in space on its own then it would remain together (except for slowly evaporating) because of surface tension. But if you let it out of the bottle with non-zero speed then there might be various instabilities and smaller balls can form.
But it all depends on the actual speed the liquid is leaving the bottle and the precise shape of the bottle (or at least its neck). Just consider showering with a hose -- lots of droplets form there.
A: The liquid does not stay in one piece, because there is an excess kinetic energy it gains when coming out of bottle, this energy is in the end transformed into the surface tension energy of the liquid drops.
One more question is as follows: why the liquid drops in space, while attracting to one another, do not assemble into a single drop? This is because the gravitational force among them is far to small for that to happen.
A: Given a stream or column of liquid, it "wants" to find some local minimum potential energy state. Less energy is required to "pinch off" a number of smaller spheres than to bring all of the water together in one large sphere, so that's what happens.
