It really depends on how much work you want to do.
Depending on the amount of storage you need, you could go with a cheap VPS from Linode or Digital Ocean (Linode has better plans IMO) and use a basic LAMP stack install to achieve this.
If you do not want to use a domain name,
this guide will explain how to install a LAMP stack on Debain 8. If you are wanting to use a domain name,
this guide should be able to assist you.
If you would rather use something a bit more front-end, I would suggest hosting with HostGator, DreamHost, or GoDaddy. Just make sure you have some form of FTP access, as well a way to create and edit MYSQL databases.