FileMaker can do all of this, but can you do it with FileMaker? That's a question you will have to answer for yourself. It will depend on your skills, experience and the amount of time and $$ you are willing to invest in learning how to set up such a database.
You can download a free 30 day trial copy of the database to see what you can do with it and there are many training resources also available--some that are free.
And we'll be glad to answer specific questions here in the forum to help you climb that learning curve...
As to the specific project that you have in mind, you can insert either the document or a reference to the document into a container field in FileMaker, then, much like a hyperlink, you can set it up so that a mouse click (or in some cases, a double click) opens a copy of the file. There are quite a few different options in how you work with a container field--including an external storage option that copies all inserted files to an external storage location with an option to make that a "secure" storage of your documents.
Security settings can be set up to limit a user's access to just the document records that they are authorized to access.