RStudio Server can be accessed via web browser after authenticating the user. If you are working in RStudio Server and mistakenly close the browser, then it will not affect your computations or saved files. You will see the same screen which you have closed after starting the RStudio Server again.
RStudio Server stores data and code files on the cloud. Whenever you will upload any new file/data or save your code in RStudio Server, then it will store those files in the cloud and not on your local drive.
Also, please note that after uploading files you will be making changes in the files which are uploaded to the cloud. Similarly, if you are using any data set in RStudio Server, then you will be using that from the cloud.
If you would like to have a copy of your work on your local drive, then download that by selecting the “More” tab (lower right window) and then click on “Export…”