Or cheat and connect the ground to the neutral. Make sure you identify the neutral at the outlet. Some old houses may be wired backwards. The neutral should be the larger flat pin. Some older outlets have both pins the same size, so use a multimeter to see which one is hot. We did this all the time in Brazil, but is definitely not up to code.
This is what I would do. The ground wire is a safety feature to keep people from being killed by electrical appliances that have internally shorted. If you follow the ground wire back to the box, it will be connected to the same neutral/ground for the house. So by attaching the ground to the neutral at the socket, you are defeating the shock safety part, but it is not a large risk, especially for something like a Tesla that has smart charging and GFCI characteristics.
Just to be clear- this is not technically legal, and is definitely not up to code. But it works, and I personally put the risk very low. You can replace the outlet when you move, it's not a permanent change.
Yeah, your only real choice is to connect a ground.
@doghousePVD's first suggestion to run an external earth ground to a stake or water pipe is what I'd probably do in a pinch.
But I'd not consider this a permanent or even semi-long-term solution. Pulling 12 amps through obviously old wiring and a cheap externally grounded 3->2 adapter for hours and hours on end is a recipe for a house fire.
I don't think that's a real concern. The only way you get a house fire with this would be if the breakers/fuses in the box are broken or the wrong rating. The entire point of fuses and breakers is to stop the current if it's too much for the house wiring. To clarify- there is ZERO current going through the ground wire. It's strictly a safety feature for scenarios where the neutral fails either in the wall or in the device.
Even if his house is knob and tube wiring, that wiring can easily handle and is specced for the 12 amp load for a Tesla. It doesn't match current code, but is not required to be replaced because it's not risky.
Another option might be to have an electrician add a 220V circuit for your charging. Even as it's a rental, this does not have to be a permanent installation, and would not necessarily cause any damage.
Depending upon your relationship with the landlord, they might allow this as a good perk for future renters, and actually encourage it, as long as they don't have to pay for it.
So my advice: Good relationship- add a 220V circuit. Bad relationship- rewire the ground on the 110.
As an aside- in my garage, I have a plug where the Tesla 110v charger worked in the top socket, but not the bottom one. Very odd. Same no-ground blinking red light. I just recently took it apart to inspect it, and, the ground for the bottom socket had been broken sometime in the past so that it no longer had the metal socket for the ground plug. Replaced outlet with a new one and now both sockets work.
Source: I'm an electrical engineer, so I've got a better than average background in electrical systems.