I'm not sure if this is a good question, but I know Azure Media Services has content protection for video and audio, and I know Azure Rights Management exists for documents and email, and seems to use a special client to view protected documents.
If I were to build a web application that lets users view sensitive documents, like CVs or financial histories, is there a way to let users view those documents (pdfs, word documents, whatever, they'd be uploaded as Azure blobs) in an ordinary web browser like Chrome etc. but without being able to download them (most importantly), print them, copy portions from them, and so forth?
Any type of content protection would need to be built by you. Blobs simply contain data that you put there.
You can make a blob private so that only your app can get to it, unless you generate a temporary Shared Access Signature (or policy). However: If you provide a link via SAS, there is no stopping someone from downloading it (until the link expires).
If you want to do something related to web-based browsing with content protection, you'd need to download the content from blob to your web app first, and then serve that content from your web app with whatever protections you wish to implement / take advantage of, without ever providing a blob's direct link to the the end-user.