Admittedly, my experience with both MySQL and PostgreSQL is limited. I spent most of my career working with Access and SQL Server. However, it is my understanding that any database server that is an actual, real "server" will take care of that issue.  

The problem with Access and corruption is that you've got multiple different workstations trying to manipulate a single ACCDB file over a sometimes unstable network. And that's why Access corrupts. It's not Access's fault. It does well what it was built to do. But it was not built to be a multi-user server.