"I have to tune it again now though, but that's not gonna be a problem".
Be careful with that and keep track of your adjustments so that you can return at least roughly to where you started. Without any test gear you are sort of working blind, and if you start adjusting more than one coil you can get lost very quickly.
Grounding is a complicated subject, but in simple terms think of you and the theremin as being part of a closed electrical circuit. It isn't necessary for different elements in this circuit to be connected by wires, as they clearly are not with a theremin. Your body is a conductor (albeit a poor one) that interacts with the oscillating electric fields surrounding the theremin's antennas. But in order for the theremin to be able to generate an electric field from its internal AC voltage source (the oscillator) it needs a fixed reference for that voltage to "push" against.
Likewise, for your conductive body to be able to affect that electric field it needs to not simply follow the voltage variations as it enters the field. Your body must not be able to simply bobble along with the voltage variations in the electrical fields, but it must tend to resist them. In one way or another your body must be held nearer to the voltage reference that the theremin is pushing against. This reference is what is usually called ground, and this term can mean different things depending on the context of the subject, which doesn't help when you're trying to understand what it is. There are many types including earth ground, local ground, chassis ground, digital ground, analog ground, etc. Just think of it as a reference against which all other voltages are measured.
Earth ground is what you get when you connect to something that can be traced back to a conductor that is buried in the dirt - water pipes, ground rods, or house electrical grounds (if your country has grounded mains). A local ground (this can go by different names) is some other means of connecting yourself back to the theremin - a wrist strap, metal chair, conductive floor mat, or a wire held in your teeth (not recommended!) any of which would be connected back to the theremin's ground side. Earth ground can be the same as the local ground, and here in the U.S. that is usually the case if you ignore noise issues. Contrary to popular belief a theremin does not need a true "earth" ground any more than a flashlight needs to be connected to a stake in the ground (though in noisy environments there may be a benefit in having an earth ground for a theremin). But like any other device the theremin does need a local ground, or a path through the player that ultimately closes the circuit.
Another way of looking at it is that the theremin can be considered a two terminal device that must have both terminals connected (not necessarily by wire) to close a circuit. The antenna is one terminal, and that side of the circuit is closed through the capacitance created by the proximity to your arm/hand. But what about the other terminal? That ideally would close the circuit by direct connection to your body. But since players generally don't like to sit in metal chairs or wear conductive straps while playing, the theremin ground-to-player connection is usually made the same way as the wireless connection through your hands to the antennas. And that's by connecting the theremin to some reference ("ground") that your body is also connected to, by proximity and capacitance.
So in summary, your body needs to be part of the circuit between your theremin's antennas and the chassis "ground" of your theremin. A connection to the dirt, whether on Earth or Mars, or the metal floor of the International Space Station isn't an essential part of the circuit.
This is a lot like trying to define a word without using that word in the description. But if you managed to get to the end of this maybe you'll have a better picture. Or not 
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