At level 10¹ you can create a Hardcore hero. My friend ask me this and I didn't know how to answer him.
If I level up my barbarian until level 10 and create a hardcore, Can I delete my soft-core barbarian and have an account with hardcore only?
Yes, you can. Just try it out: Create a level 1 hardcore dummy, then try to delete your non-hardcore character. Nothing will stop you from doing so.