Is it possible to store files on Ubuntu One without syncing them to a (or every) computer?

Yes, it's possible.

While files that are placed in the main Ubuntu One folder (~/Ubuntu One/) will currently get synced to all your Ubuntu One enabled computers, you do control which additional folders get synced. Although this functionality isn't currently exposed in a user-friendly way, you can still use it via u1sdtool: you can subscribe (and unsubscribe) to folders using u1sdtool --subscribe=<folder id> (and, respectively, --unsubscribe=<folder id>); you can get the folder id from u1sdtool --list-folders.

When on one computer you select “synchronize this folder with Ubuntu One” in nautilus (or u1sdtool --create-folder=/path/to/folder), your other computers do a “folder merge” and start syncing that folder too, if the value of udf_autosubscribe in /etc/xdg/ubuntuone/syncdaemon.conf is set to True.

So, on the computer with a smaller hard drive, edit that file and set the value of udf_autosubscribe to False. You can then manually add the folders you want to be included using u1sdtool. With this you have access to some of your files immediately, and you can fall back to the web interface for the rest.

This is not the same as having something like a webdav mount, but it probably covers your needs.

It is possible in the latest versions to disable what folders are synced to a particular computer by unchecking them in the folders tab. Note that they will still be stored on the cloud for all other computers.

This won't automatically unsubscribe after so much being downloaded, but you have more control over what is selected, and it's a halibut easier.

Further, there is an option to not subscribe automatically to new folders. Try that, too.