So, it's just like the title states. I try to save in game, and a message says "Not enough hardspace to save". Any fix for that?

How much free space you have on disk where SS1 is installed? The EE is windows executable? (Didn't verify). If you were playing the classic under DOS (dosbox), I would expect it to have problems with certain amounts of free space (too much), but whether the windows enhanced edition may have such problem, that would be a bit sloppy programming.

Also the error message may be just misleading for another kind of failure, like writing rights not being granted, then trying out to run it as admin may help override that.

This is just general hint post, not directly related to any experience with SS1 EE (IIRC I didn't ever tried EE myself, nor do I have windows OS anywhere, so I'm not capable to help with any details).