Any solution will be "technical" in nature since the problem involves technology anyway. However, for the "access" issue, you can create a solution which puts more work on you than on the technology. (Since tech is supposed to make human work easier, that's kind of an anti-tech concept, right?)
Required environment is a publicly (relative to the classroom computers) accessible file storage in which you (the instructor) can create password-protected directories. [This can still be a closed environment relative to the wider network, including other parts of the LAN, and the Internet as a whole, so long as it is a location that the students can access during class.] Out-of-class access could be an issue, either because it is desirable and making it happen is not easy on the school's system, or because it is undesirable, yet hard to prevent on the school's system. As mentioned in another answer, the students can always print a copy, or email a copy to themselves, for out-of-class review and planning.
For the project, create a directory for each pair, or team of any size, with a unique (per directory) password. Provide that password to each of the team members. When one is absent, for any reason, the other (rest) can still access the directory. If you have the need to reassign team membership, you change the password(s) in the affected directories and give them to the reorganized teams.
This provides each team a space to work, and share, including a place to make related notes, trials, documents, etc. You, as the instructor, will have access to all the directories, and can check the work at any time. The solution works equally well for program coding, research paper writing, and even graphic design. The hard part for the instructor is creating, and maintaining, the collection of directories and passwords. Especially if there are overlapping project time-frames, requiring that there be multiple sets of directories available at one time (along with remembering who has access to which one) with each directory needing a unique, yet easily used, password.
This solution does not scale well at all, and is unworkable in a workplace environment, yet is simple enough to set up and use in the controlled environment of an educational arena. At the end of the term you can merely delete/clean the parent directory and you're ready to start the next term.
Of course, the underlying principle may be enhanced, or modified, using ACLs, Apache user names, SSH with PGP keys, and many other "technological" variations; but, non-technological was the target.