We can set you up with SQL Server, a robust, scalable database solution.
SQL Server Express is a free version of SQL Server which is a great solution for an in-house database. All you need is a computer to host it on.
Microsoft Azure is a 'Cloud' version of SQL Server. A small cloud-hosted database (less than 2GB) can cost as little as $10 per month, and is able to be upsized at any time without any interruption to workflow.
Azure includes a continuous backup and restore facility which is set and forget. This is a great way to get your database online, without the headaches of owning, securing and maintaining your own server.