Is it somehow possible to extend one's webspace by connecting to a server which has more space?
Lets say I have a remote server with a small disk, and a local server with plenty of free disk space. Can I somehow permanently mount the space on my local server on my remote server?
Sure, you can use something like sshfs, but this seems to be a ridiculously complicated and fraught way to deal with disk space issues. It's a solution you'd use if you didn't care about uptime on your webserver and the effort you'll spend dealing with it when things break.
A better way would be to pay your hosting provider to get more disk space. Another way would be to redesign your website to use more proper offsite storage, like Amazon's S3.
iSCSI is one common way to do this. Another would be NFS.
Both are not a good fit for a WAN scenario, especially when a small bandwidth network is included, for latency and security reasons.
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".