You could extend git-lfs "pointer" file to support secure distributed storage using Convergent Encryption [1]. Right now, it's 3 lines:
version https://git-lfs.github.com/spec/v1
oid sha256:4d7a214614ab2935c943f9e0ff69d22eadbb8f32b1258daaa5e2ca24d17e2393
size 12345
By adding an extra line containing the CHK-SHA256 (Content Hash Key), you could use a distributed p2p network like Freenet to store the large files, while keeping the data secure from other users (who don't have the OID).
version https://git-lfs.github.com/spec/v2proposed
oid sha256:4d7a214614ab2935c943f9e0ff69d22eadbb8f32b1258daaa5e2ca24d17e2393
chk sha256:8f32b12a5943f9e0ff658daa9d22eae2ca24d17e23934d7a214614ab2935cdbb
size 12345
That's how Freenet / Tahoe-LAFS / GnuNET work, basically.
[1] https://en.wikipedia.org/wiki/Convergent_encryption