EDIT: According to Jonathan of RStudio a new version doesn't need libgstreamer. See his post below for the relevant link.
RStudio won't install on Ubuntu 16.10 as key software inputs - libgstreamer0.10-0_0.10.36-1.5_amd64.deb - can't be installed. It seems to be obsolete, the system refuses to install it, and I'm unable to find an alternative.
There are suggestions to install the missing software alongside libgstreamer-plugins-base0.10-0_0.10.36-2_amd64.deb, but only the latter works on my system. I tried apt-get to download it directly, and also got the package for use with dpkg or gdebi. The Ubuntu Software GUI doesn't seem useful, as it doesn't install despite giving no error message.
As the answers provided are insufficient in my case, the issues still appears to be unresolved for some of us.
RStudio has a new preview out that doesn't require libgstreamer0.1 (it uses the new libgstreamer1.0 in Ubuntu 16). You can try it here:
https://www.rstudio.com/products/rstudio/download/preview/
This will become the stable build in the next few weeks.
Turns out that it IS possible. Thanks to Mike Williamson for excellent instructions, reproduced below:
1) Get the latest R Studio Daily Build here, though note that it's not necessarily stable.
2) Install, chaning the name of the package to the one you downloaded - perhaps easiest if you go to your Downloads directory - and you'll probably find that there are missing packages:
sudo dpkg -i rstudio-1.0.124-amd64.deb
3) Download the missing packages (the lack of which causes the installation to fail):
wget http://ftp.ca.debian.org/debian/pool/main/g/gstreamer0.10/libgstreamer0.10-0_0.10.36-1.5_amd64.deb
wget http://ftp.ca.debian.org/debian/pool/main/g/gst-plugins-base0.10/libgstreamer-plugins-base0.10-0_0.10.36-2_amd64.deb
4) Install them:
sudo dpkg -i libgstreamer0.10-0_0.10.36-1.5_amd64.deb
sudo dpkg -i libgstreamer-plugins-base0.10-0_0.10.36-2_amd64.deb
5) Make sure they don't get over-written at the next software update:
sudo apt-mark hold libgstreamer-plugins-base0.10-0
sudo apt-mark hold libgstreamer0.10
6) Install RStudio (changing name to the version you downloaded):
sudo gdebi rstudio-1.1.5-amd64.deb
7) Launch RStudio:
rstudio