If you were doing this then you would use RDP to connect to it more than likely and then a VPN for people out of the house. That said, beyond the fun of setting this up, it seems like it would just be impractical compared to everyone having their own local machine. You aren't going to get decent graphical performance if anyone wants to run games and doing anything like watching video is going to stress the machine and the network a fair bit. This is going to be even more of an issue for the people who are located outside of the house as you'll be pretty limited by the internet connection.
As far as the software goes, you can't use Windows 10 as it'll only allow one remote desktop connection at a time. You would need a copy of Windows Server (£600) and strictly speaking you'd also need the appropriate "Client Access Licences" for each user which would work out to be a further £600 for 5 users. Office 365 is fine as long as each user has their own subscription.
While it is a fun project to set something like this up, I wouldn't spend a lot of money on it and plan for everyone to use it, it just isn't that practical for regular family workloads. I personally like experimenting with this sort of enterprise level networking stuff at home, however I keep it pretty separate from stuff that I actually need to do work! Remote desktop systems like this are common in some business environments but those are usually for places like call centres and offices where staff only run a couple of pretty boring applications all day. They do not work well when people start to do stuff like watch video or try to play games, even audio can be tricky. You'll also run into silly problems like when someone wants to connect a USB device or use a webcam, this really doesn't work well with RDP systems.
My advice would be, if you want to do this so you can play with the technology and learn how it all works, grab an old PC or pick up a cheap used server and have a go. It can be good fun! However, I wouldn't recommend pushing people onto a system like this for their day to day workloads, you'll just end up having a nightmare trying to support the whole thing!
