Why this restriction: "I want to ensure that I don't have two people working on the same database at the same time"?
You could run the database using one copy of FM Pro, and the others can log on using their web browser.
What Sorbsbuster is describing is called "web publishing" and is usually accomplished with "Instant Web Publishing" (IWP)--something for which there's a PDF document included with FileMaker that you can read to learn more. You can find this PDF in the documents submenu in the FileMaker Help menu.
It's the least expensive option in terms of $$, but can require an added investment in time and testing to make sure that layouts etc that you design for your database work with the web browsers used to access your database. The next step up in cost is to have each of your users acquire FileMaker Pro and use remote access to connect to your hosted database instead. This allows for less limitations to your interface design, but requires that added cost--so it's up to you. Neither approach requires FileMaker Server as long as the number of users are small. (up to 5 at once time with IWP, up to 9 at one time with FileMaker Pro.) There are also hosting companies that "rent" out the use of their copies of FileMaker Server for a monthly fee.
Sorbsbuster asks a very key question about that use of the term "database".
In fileMaker systems, sometimes "database" refers to a single table. Sometimes it refers to a single file with many tables. Sometimes it refers to a system of many files--so it is important to clarify if you choose to use that term when posting a question here in the forum.