Sure, there have been agencies doing this for years. I’ve used several of them over the years when I was visiting. The thing you’re looking for is called a Mitwohnzentrale, which is sort of like a roommate-finding bureau, but also does short-term sublets. I don’t currently know of any, although there seems to be one a couple of blocks from me. The woman I used to use went out of business years ago.

If you’re planning a stay of a few weeks, it beats the pants off of a hotel, since it’s not only cheaper, but you wind up living in a neighborhood and having to shop and get in and out just like your neighbors. Be aware that the phone is usually turned off, and the internet connection (and possibly even the cable) could be, too. Work all this out with the folks at the MWZ.

Germans get 6 weeks holiday, for the most part, and tend to take three in the winter and three in the summer, but some folks take all six at once. Figure out when you want to come, then Google Mitwohnzentrale Berlin and see what you find.