Starting an application where a server is needed, but don't have one.

Not sure if this is the best sub, but here it goes.

For my application, sparing the details, I need a database on a server. I was going to build one just for small scale testing, but my question is, for a small startup, what would be the best option when a DB server is needed.

Is it possible to rent or buy server space somewhere? I'm perfectly fine coding this thing, but maintaining a server of my own is beyond my knowledge at this point. Especially the cost and connection logistics.