Xbox 360 on carpet: did I damage it?

The Carpet is a bad place for an Console. If you have no Problem in the meantime, than it is luckily alright

Static build up can also happen so you should be good if noticed no issues just make sure you keep it off the floor and well ventilated.

Hi, whilst placing your Xbox on any soft surface such as carpet isn't advised I doubt you have anything to worry about. The Xbox would have overheated and shut down no doubt had there been a problem with air flow.

I have an Xbox 360 S and the vent is on the top. I had it on the carpet for a few days, not knowing this was bad. Could I have done any damage to my Xbox? I haven't noticed anything.

If you haven't experienced any issues (heating, shutdowns, etc.), I doubt any damage was caused. Although, I'd recommend placing your console elevated off the floor, especially carpet as it tends to harbor excess dust/lint/debris which could cause ventilation
issues.

I can say pretty definitively that the likelihood of you doing long term damage to your console just from it being on the carpet for a few days is extremely low. That said, ensuring proper ventilation and heat-exchange is super important to not cutting your
console’s life expectancy. Someone above posted the article on ventilation for the Xbox One, this article is specifically about optimal placement of your Xbox 360. Cheers!

https://support.xbox.com/xbox-360/console/best-location-xbox-360