Its possible, and decoupling storage from the server is a reasonable scenario to connect a server to a shared, highly available, storage device (e.g. a NAS or SAN) as part of your initial design for that server.
I wouldn't "just add some space from another server" to a server that was already deployed, no. Instead of your web service being dependant on one server and its link to the Internet, you'd create a service that was dependant on your link to the Internet, both servers, and any connections between those two servers. You'll have made your service inherently much more unreliable just for the sake of saving a bit of money adding storage to the web server in a more reasonable way. This would be a good demonstration of the British saying "Penny wise, but pound foolish".