Can pressure exist without a container?

The pressure of a gas is defined as the force the gas would exert upon a surface or container. However, there is no need for a container for pressure to exist. For instance, the air you're breathing right now (unless you're in an airplane or submarine) has pressure due to the column of atmosphere above you. Stars are balls of gas (plasma, actually) that are pressurized by gravity; no containers to be seen.

An addition to Daniel's answer:

It regularly happens that a certain definition isn't applicable when talking about liquids and gases. For example, almost all definitions in thermodynamics are defined for closed systems, which is rarely applicable in the real world.

To overcome this, there is the notion of a control volume. For this, we basically take a slice out of our medium (be it gas, liquid or anything else). We put this slice in a special box - so special, that it behaves exactly like the medium that originally surrounded our control volume. We can now forget about the rest of the medium, and just focus on what it does to the walls (boundary) of our box, and what our control volume does to the walls.

Related to your question, we can now take a little slice out of the atmosphere, and put it in our box. We can now forget about the rest of the atmosphere, i.e., put the box in a vacuum. Intuitively, you would think that the box should blow up like a balloon when put in a vacuum. However, the box behaves like there is an atmosphere, and will retain its shape - it will create a force on its walls. Now, we see pressure arise clear as the day: there's a force on the walls of the box, and since the walls have an area, you get p=F/A.