I tried to find an existing bug for this, but surprisingly, I wasn't able to find one. We need this especially for Notifications/Echo.
Currently there is Extension:ShortUrl, but the URLs generated by this aren't much shorter than the existing URLs.
We also can't use most 3rd party services due to privacy concerns.
The main thing we need is a short domain name. This would most likely have to be donated or granted to us, as short domain names are not cheap.