Using Dropbox to Sync File GeoDatabases?

ESRI intentionally obfuscated the FileGDB file structure so people wouldn't try and mess with it internally:

A file geodatabase is stored as a folder of files. When you view the folder with Windows Explorer, apart from the .gdb extension, it looks like any other folder and you can view its contents. The folder contains cryptically named files that hold geographic data, attribute data, index files, .lock files, and a signature file, as well as other files. Each feature class or table in the geodatabase is stored in two or more files. By design, it is difficult, if not impossible, to tell which files make up any given dataset.

ESRI recommends that you copy FileGDB's with their tools and not your OS tools. However, it is possible to copy the whole database or rename it:

Although not recommended, it is possible to copy a file geodatabase to another location by copying the folder to another location (if you are using a Windows Vista operating system, you also need to change the extension of the new folder to .gdb; otherwise, ArcGIS cannot recognize it as a file geodatabase). Only copy a file geodatabase with Windows Explorer when you are certain there are no other processes accessing the data. The problem is that it can be difficult to tell whether other processes are accessing the data. You might think there are no other processes; however, a process on your computer or another one could be accessing it. If you happen to copy the folder when another process is accessing the data, the data may not copy properly. An improper copy operation is often not apparent—you might not notice any problems until you access a certain part of the new copy a few days later.

I would be wary of using dropbox to sync the files individually. The database structure just isn't built to be manipulated in that way. If you dataset is small enough, it may be a good application for a personal geodatabase or a sqlite database (though I haven't seen it confirmed that ArcGIS can write sqlite) if you have simple geometry.

There is a tool that will recover corrupted geodatabase in the event they are damaged:

If you need to have multiple PCs accessing files, it would seem to make more sense to migrate to a true spatial server setup rather than an ad hoc one using Dropbox. This could be either a hosted server running ArcGIS or a PostGIS database running on a cloud web service like AWS. My preference would be for the latter. It'll take a little more work to set up, but you can define your own relations and scale up to as many computers as you need (depending on how you configure AWS), all for much less than you'd be paying Esri.

There are several SO posts that discuss PostGIS and ArcGIS here and here. The second one seems to have more up-to-date information in the comments below the accepted answer.

Without something more robust handling the traffic from the different clients, you're probably going to be constantly dealing with data corruption issues. At the very least, if you moved to shapefiles, you should check out GeoGit. This would give you basic revision control so shapefiles edited on one computer can be merged with edits from another without being completely overwritten.