It is possible to create a database with the MDF and LDF files on a network share (using a UNC path or a mapped network drive) and run a database from a network location even if the service is running on a different machine. Although MSSQL itself cannot see mapped drives or UNC paths, a Windows user can. To create a database with MDF and LDF files in a network location, you must create the database first, specify the file locations, and then restore the database over it (with a script).
Note: There may be a way to specify during a database restore (with a script) where to put the LDF and MDF files, but that has not been tested.