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What would I feel if I went swimming in a ball of water in space? Would I feel greater pressure as I went deeper into the sphere? What would it be like to swim in something like that?
Also, let's say that I don't need air to survive in this scenario.
Best answer is given from xkcd's what-if: https://what-if.xkcd.com/124/. It's not space, but it describes the fluid flow in lower gravity - such as how you could jump out of the pool just by performing aquadynamic maneuvers, or walk on the water. It is really a cool read.
As mentioned in the xkcd article - diving and floating, being primarily about differences of density and viscosity, wouldn't change - you'd be able to dive at about the same speed as before.
Due to the law of gravity, the ball of water would have some gravity, simply due to it's mass. By Newton's shell theorem, you would have maximum gravitational attraction near the surface, and while the pressure would go up as you dived down (proportional to the ball of water's gravity), the gravity would go down. However, this effect isn't very measurable (if you want to see why, keep reading - otherwise skip to the bottom).
Assuming an unprotected body, you wouldn't be able to go very deep, or you wouldn't have a ton of water to hold this ball of water together - you really can only go down about 50m - 100m, according to the world records
A 100m radius ball of water, as Wolfram points out, is only 3 thousandths of 1% of earth's gravity - essentially nothing. Wading through this ball would be pretty much like spacewalking, but with inertial drag to stop you.
Scale the radius up by 100000 times, and you'll feel 30% of earth's gravity, (and your ball of water would stay together) but you would feel none of the variation with depth, since you can't dive very deep.
The breaking point between these two points would be when your swim speed would reach the escape velocity of your ball of water. In this case, Wolfram helps me again to show that would be around a 2.68 km radius. Smaller than that, and when you swam out of the ball, you'd just float away. Bigger than that, and the ball would catch you and pull you back in.
So, the bottom line is swimming in a big ball of water pretty much feels like swimming very slowly in space - until the ball of water gets big enough (2.68 km). Then it just feels like swimming in a giant pool on a distant planet. For practicality, the ball of water doesn't work, but the lunar swimming pool is awesome.
Edited for clarity on the source of the gravity, while the poster mentioned 0g.
Let’s assume that by “swimming in space”, it is meant that there is no gravity whatsoever. A (basic) microscopic view of the pressure as the force exerted by the particles (e.g., molecules) of a fluid on a surface implies that still there will be a pressure on a macroscopic body in the fluid, regardless of external forces, like gravity — although in a static situation, the net force on the body will be zero.
If outside of the fluid, the pressure is zero, then it means if you leave the surface and go into the bulk (inside the fluid), you will feel more pressure. But the pressure will be (almost) the same throughout the bulk of the fluid.
Finally, remind that what I said above holds true only in a quasi-static case. As one swims through the fluid, one perturbs it a lot, and this means local pressure will be non-homogeneous in space and changing with time.
