Tente instalar o OwnCloud utilizando SSL / TLS, ele realmente usa o WebDAV e está em conformidade com todos os seus requisitos.
OwnCloud (stylized ownCloud) is a suite of client-server software for creating file hosting services and using them. OwnCloud is functionally very similar to the widely used Dropbox, with the primary functional difference being that OwnCloud is free and open-source, and thereby allows anyone to install and operate it without charge on a private server, with no limits on storage space (except for disk capacity or account quota) or the number of connected clients.
OwnCloud deve ser configurado para usar criptografia SSL / TLS
Tem clientes legais para Android, OS X, Windows e Linux.