RX300 won't turn off after removing key...

[Mystery123]

Last year, it happened once.

I took the key out and the engine didn't turn off.
It was still running. I tried several times putting the key in/out but it still kept running.
I drove it around and after driving little bit, it finally turned off.

It worked fine for almost a year and it happened again. 
This time, same thing, many tries on using key in/out did nothing, the engine still was running so I drove it around but it died while driving.

I smelt something burning on the exhaust. Now I can't get it to start back again.
Thankfully, it's on a parking lot currently. I'll try to get a jump start later and see if that works.

What do you guys think? What's happening?

 

 

[Mystery123]

Update:

I charged the battery and installed it.

As soon as I connected the terminals, engine started right there so obviously it's not the battery.

Also, I saw smoke on the +ve terminal wires so I took the battery as soon as I could.

I towed it to a repair shop nearby. They called me today that they found the issue. It is starter and needs replacement.

Close to $500 total repair cost. 

Maybe time to trade it in after this repair as other things will start failing, after all it's 18 years old.
This vehicle only is worth less than $3k anyways and trade in would be probably less than 2k.