The only way to do EXACTLY what you want to do would require a server solution making use of Open Directory. Open Directory is essentially a service that store user information and manages policys and permissions on your network. This solution is exactly what you want, but is complicated and expensive.
If you are simply trying to enable redundant file sharing across multiple computers you should invest in a network shared external hard drive.
http://www.newegg.com/Product/Product.aspx?Item=N82E16822136346
This hard drive would be accessible from any computer on your network. You can simply make folders on the drive with the users name.
Let me know if you have any other questions.