My Pi3 runs two applications: The UrBackup network backup server that I already mentioned, and also NextCloud server. NextCloud is like DropBox, iCloud, etc. except you run it yourself rather than paying for someone else to provide your cloud. NextCloud is free (as is UrBackup). You can sync Linux, Windows, Android, iOS with it. Maybe others too, but those are the only ones I've tried.

A Pi3 runs both of these applications, simultaneously, very well. Just get yourself an external hard disk for storage. I bought a Western Digital Red 4Tb drive because I wanted something that runs cool, quiet, vibration free and was rated for 24x7 operation. I bought a separate external enclosure - solid metal construction with a cooling fan. I have since found that the WD Red drive runs so cool that I have disabled the enclosure fan. No need for it. The drive never feels any warmer than ambient temperature, even during long accesses.

The one thing about a Pi3 though, is it only supports USB2.0 currently. People are speculating (and hoping!) that the next generation Pi4 might support USB3.0, but nobody knows. The Pi developers keep future enhancements close to chest. I find the current USB2.0 to be perfectly adequate for my Backup and Cloud needs. But obviously, USB3.0 would make those applications even faster.