While running batch scripts is certainly a good option, it's not the only solution. Sometimes you may still want interactive use in different sessions rather than having to do everything as batch scripts.
Nothing stops you from running multiple instances of RStudio server on your Ubuntu server on different ports. (I find this particularly easy to do by launching RStudio through docker, as outlined here. Because an instance will keep running even when you close the browser window, you can easily launch several instances and switch between them. You'll just have to login again when you switch.
Unfortunately, RStudio-server still prevents you having multiple instances open in the browser at the same time (see the help forum). This is not a big issue as you just have to log in again, but you can work around it by using different browsers.
EDIT: Multiple instances are fine, as long as they are not on the same browser, same browser-user AND on the same IP address. e.g. a session on 127.0.0.1 and another on 0.0.0.0 would be fine. More importantly, the instances keep on running even if they are not 'open', so this really isn't a problem. The only thing to note about this is you would have to log back in to access the instance.
As for projects, you'll see you can switch between projects using the 'projects' button on the top right, but while this will preserve your other sessions I do not think the it actually supports simultaneous code execution. You need multiple instances of the R environment running to actually do that.
UPDATE 2020 Okay, it's now 2020 and there's lots of ways to do this.
For running scripts or functions in a new R environment, check out:
the callr package
The RStudio jobs panel
Run new R sessions or scripts from one or more terminal sessions in the RStudio terminal panel
Log out and log in to the RStudio-server as a different user (requires multiple users to be set up in the container, obviously not a good workflow for a single user but just noting that many different users can access the same RStudio server instance no problem.
Of course, spinning up multiple docker sessions on different ports is still a good option as well. Note that many of the ways listed above still do not allow you to restart the main R session, which prevents you from reloading installed packages, switching between projects, etc, which is clearly not ideal. I think it would be fantastic if switching between projects in an RStudio (server) session would allow jobs in the previously active project to keep running in the background, but have no idea if that's in the cards for the open source version.