I have a way to do it, using symlinks so that Evolution knows where the files are, but they are also sitting in my Dropbox folder. You have to re-create these symlinks every time you make a new, clean install of your OS (in my case, Xubuntu), but the small effort is worth it. Caution! Do not open Evolution in a clean install until your hosting service has made a complete sync to your PC! Or, use Evolution's Backup/Restore function to restore all of your e-mail files AND setting up symlinks (as shown below) BEFORE connecting to Dropbox (or whatever) after a clean install.
Before sync'ing Evolution on a NEW installation of OS, copy the account info from your previous installation, or use values from another PC. If you don't do this, you may find that you can't access Contacts, or may encounter other aberrations.
If you're only doing an upgrade from a previous version (without new install), you probably won't have any problems.
For sync'ing Evolution, note that there are 3 Evolution folders that have to be copy/pasted and then linked to.
Before starting my procedure below, install the Evolution e-mail client program. Then, read the instructions below entirely, before starting the procedure.
1st --
*For a preexisting install of Evolution, copy the existing "evolution" folder in ~/.local/share, and paste it into ~/Dropbox.
*For a new install of Evolution, just create an "evolution" sub-folder in the Dropbox folder.
For either situation, re-name the existing "evolution" folder in ~/.local/share as "evolutionX". Create a symlink to the "evolution" folder in Dropbox, then cut and paste the link into ~/.local/share.
2nd --
*For a preexisting install of Evolution, copy the existing "evolution" folder in ~/.config, create a sub-folder in Dropbox, "~/Dropbox/Sync_folders" then paste it into ~/Dropbox/Sync_folders.
*For a new install of Evolution, just create an "evolution" sub-folder in ~/Dropbox/Sync_folders.
For either situation, re-name the existing "evolution" folder in ~/.config as "evolutionX". Create a symlink to the "evolution" folder of ~/Dropbox/Sync_folders/evolution, and cut and paste this symlink into ~/.config.
3rd --
*For a preexisting install of Evolution, copy the existing "evolution" folder in ~/.cache, then create a sub-folder in Dropbox as "~/Dropbox/evolution_instructions" and paste it into "~/Dropbox/evolution_instructions".
*For a new install of Evolution, just create a sub-folder, "~/Dropbox/evolution_instructions/evolution".
For either situation, re-name the existing "evolution" folder in ~/.cache as "evolutionX". Create a symlink to the "evolution" folder of ~/Dropbox/evolution_instructions/evolution, and cut and paste this symlink into ~/.cache.
If you have have installed Evolution in a NEW system install, and if you never opened the Evolution program, there MAY be no existing "evolution" folders to rename with "X" at the end. In that case, just paste your symlinks and make sure they are re-named as "evolution".
Renaming -- By renaming all of the pre-existing "evolution" folders as "evolutionX", you retain all of the pre-existing folders under a revised name -- they are available in case you want to go back to the status before you linked to the folders that are synced to Dropbox.
Cut and paste these instructions into a text file, named "Evolution instructions for syncing e-mail.txt", and place this .txt file into the ~/Dropbox/evolution instructions folder. That way you will have this information for a future "clean" install of your OS.
The reason for the separation between "Dropbox" and "Sync folders" and also using the "evolution instructions" folder in "Dropbox" is that these folders are all named "evolution", so you can't have 2 or 3 sub-folders with the same name in the same higher folder. Of course, Evolution normally handles this by using 3 separate higher folders in the Home folder. You have to maintain that separation in parent folders.