After seeing a question about alternatives to Dropbox, I thought it might be a good idea to share the few options that came up, plus an additional one that I do use myself.
One suggested option was Seafile, “an open source file sync&share solution designed for high reliability, performance and productivity.” It looks great and the community edition can be installed on your own server for free. The Pro version is free for up to 3 users and above that offers a variety of pricing options.
Nextcloud, “The self-hosted productivity platform that keeps you in control” is completely free and can be hosted on your own server. There are a few voices in this thread who recommend Nextcloud. It's the most comprehensive solution of these three.
I currently use Syncthing as a sharing tool with the Colloq team. This option is a little bit different, since it only does peer to peer sharing with dedicated machines that you allow. With this process there is no server in between and this option is great for sharing and exchanging files, but not really suited for the heavy-weight job of storing stuff on a server. It also only synchs when machines are actually online, so it does have its limitations, but to share files between a team it does a good job.
I hope you’ll find the best solution that works best for you.