Storing photos for back up and making them available anywhere to any devices, or through a browser, can be done using any of the standard cloud storage solutions.
There are some options for free storage of unlimited photos, with some restrictions on file size.
Google Photos
Google Photos is part of Google Drive, but can also be accessed easily directly with any Google account. If photo back up and sharing from smart phones and standard cameras is your main concern then this is an ideal solution. Unlimited storage as long as you keep below the file size limits.
Flickr
Available through a free Yahoo account Flickr is the best known photos sharing platform to date. There are no file size limits and it gives 1TB of free photo storage.