IT Advice around cloud

I'm not sure if anyone can help with this. We currently have a server and are looking to dispense with it, we have considered moving our files to Google Drive, Dropbox or a similar system, however, we have some bookkeeping and payroll files (with files extensions non Microsoft etc.) that will always give rise to conflicted copies if two people accidentally access the files concurrently.

Obviously with a client/server setup there can be no concurrent access, therefore there can be no conflicted copies. We are looking for a solution to this, however, all we can find so far is to use a system like box where you can lock the files - but that is not ideal, or do actually move the files to an alternative folder when in use, which is not ideal either as if someone forgets, we run the risk of conflicted copies and losing our data.

Conflicted copies cannot easily be resolved as if two different people concurrently enter data into these systems and we end up with conflicted copies, we have no way to recombine the data from each of the copies.

If anyone has a solution for us we would love to get your help to implement this.

Budget is just an estimate as we have no idea of what is required and what the time involved would be.