I'm building a fancy smart electricity meter, and I'm using one of those cheap Chinese SCT-013 current transformer clamps to monitor a live 230VAC mains wire. If I understand correctly, the primary "winding" of the CT is the wire that I clamp it on, and the secondary is a built in coil. So the winding ratio is my case is 1:1800. The peak voltage of RMS 230V is about ~325 volts. If I multiply that by 1800, I get 585,000 volts peak on the output of the current transformer. Is this really possible, or am I misunderstanding something here? The burden resistor on the CT's output is just a small SMD resistor. Is there actually half a million volts flowing through that tiny little resistor?
In theory, yes, but only if it were to measure a 230V on the short piece of wire passing through the hole of the secondary coil, which is not possible for your setup.
In practice, the primary is just a piece of thick wire, so unless your CT is used in the power grid industry, in a domestic circuit there will never be enough current to sustain a 230V drop on the primary wire.
Also in practice, never let a CT's secondary winding as open circuit. That can pop/explode your CT, or at least pierce the isolation between turns, and ruin your CT and the attached measuring instruments.
Another thing to take care is that coils (so transformers, too) act like energy reservoirs, just like capacitors. Therefore, if not operated properly (or if something fails/breaks) they can release all the stored energy very fast, so at huge powers that can do real damage.