Would it be possible to use an arduino as an internet radio station that plays the sound picked up by a microphone?
Yes in theory, but...
Which Arduino?
Which protocol as there are many?
What quality do you expect?
How many listeners do you want to serve simultaneously?
In practice it will not be easy (and that is an understatement)....
No in theory, but...
If you can do with a very low sample rate, and if there is only one listener, perhaps is can be done. But I doubt if it will be reliable.