Apparently, once the ignition is enabled, you can drive off without needing the key "nearby" (interesting experiment, I'll have to try that next time I get a "keyless entry" rental). Once you can drive off, the car is pwned - you can strip it for parts (chop shop) or fully defeat the key/ignition security away from prying eyes.
[1] You usually need to press the button a second time to unlock the remaining doors. There's also often a separate button to independently unlock the boot. A typical procedure is therefore: open boot with the relevant button, put shopping inside, close boot, take trolley back to the stand, walk back to the car, unlock only the driver's side, get in, lock door with button from the inside, drive off.
If you turn the ignition off and it can't "see" the key, it will give you a warning message on the display cluster and you have something like 20 seconds to re-start the car before you're locked out.
Also, once you've driven the car away, you'll have to find another way of starting it in future; relaying still needs some kind of physical access (albeit at 8 metres distance) to the original key.
(I'm aware of broken crypto in some wireless keys, which is purely an implementation issue - you can make these secure. That's not what this article is about. Quote: The attack works no matter what cryptography and protocols the key and car use to communicate with each other.)