Dropbox is one of the most popular cloud storage services available. It offers a HIPAA-compliant secure storage option, and while HIPAA is a US-based standard for protecting patient privacy, it is viewed by many around the world as the gold standard in patient privacy regulation. Please see the HIPAA/HITECH section of the Standards and regulations page on their site. See their "Getting Started with HIPAA" guide and Help Center article for more detailed information. Dropbox is, however, a US company and, hence, raises the issue of data sovereignty.
The "Dropbox Basic" free plan, may be all you need. That gives you 2GB of storage which should be enough for roughly 4,000 photos. If you require more storage space, you can upgrade to various paid plans.