Server at home?

Any way that I could get a server at my house were each computer had a logon screen like the included picture?

Were I work we have a username and password that we can use on any of their computers and all your files are kept on their server in my H: Drive so any computer I login on has all my files just like i left them on another computer (its like a virtual flashdrive, with 20TB per person). I want something like this at my house. I have 34 Computers and believe me, it is a total pain to transfer over the network. Want my files at home to be like my H: Drive.

So lets say I have a database on Computer 1 (We'll call it C1). Just so happens this Database is almost 300 GB. I dont have a flash drive big enough to put it on, and my current network couldnt transfer that with an ethernet cable in 2 days. If this file was at work, I could put this file on my H: Drive within a few seconds.

I think my school had a system like this, we had I: Drives and if we worked on a word document, but didnt quite finish, I could save this to my I: Drive and when I came back the next day and logged in on another computer with my username and password, my files were right where I left them on the I: Drive and I could continue working on them.

How would i do this? What kind of server would i need? How much would I spend?