Mechanically stuck.
I’ve had the same thing happen to worn out slot loading DVD drives in the past but this one has had minimal use.
Luckily the Mac is pretty much limitted to facebooking and web surfing so it’s not a huge problem, it came up because I bought a new printer this weekend and had to load a driver.
There are always the outlying mechanical lemon drives that get by the inspections. How close are you to an Apple Store? And did you get the extended warranty with your notebook? If not, you can attempt to replace the drive yourself. It would be less expensive. Here is are Instructions for how to remove the drive:
Replacing MacBook Superdrive YouTube.
Replacing MacBook Pro Superdrive YouTube.
Replacement Superdrive source.
However, it may be as simple as a loose screw and the drive being lifted a bit from the position it should be in to align with the slot in the case. I'd check that first, before ordering the replacement drive.
Oh, the reason I mentioned the Apple Store... You might want to try them. They will often fix these things free, even on out of warranty computers. You can never know... they might also charge you. It's sort of like Russian Roulette.
A friend took his two year old, out of warranty Notebook Mac in with a problem... they took it into the back and after an hour came back and said "We can't fix it here... and it would take several weeks of turn around to send it where it can be fixed. So the manager said to give you a new one." Handed him his old computer and a brand new MacBook Pro... already configured with his files transferred, and said, "Have a nice day!"