There's a lot of options when it comes to storing data somewhere on the internet, giving access from computers and smartphones wherever you are.
There's three essential different solutions:
A: Free ( commercial cloud storage
There's a lot - the most well known are Microsofts OneDrive, Apples iCloud, Google Docs, Dropbox and Evernote. Some are plain and simple storage in the cloud and some (like Evernote and Google Docs) come with special integrated applications for various purposes.
B: Privately hosted
You can set up your own storage and make it accessible from the internet. Typically in the form of a NAS and the appropriate firewall setting, that takes care of the access from the outside. When allowing "someone" (even yourself) to get access to the inside of your network you might also consider to use a VPN setup.
C: Hosted private storage solution
Instead of having to deal with the networks setup and having a personal always on computer, you might deploy a storage solution at a server hosting site. The hosting site more or less provides you with an empty Linux computer and you have pick and install a solution that answers to your requirements. Common systems are FreeNAS, ownCloud and NextCloud