How do I place blocks out of blockspaces?
The video you have linked does not use actual blocks, but invisible ArmorStands wearing blocks on their head:
You can summon one of these as such:
summon ArmorStand ~ ~1 ~ {Invisible:1,Invulnerable:1,NoGravity:1,ArmorItems:[{},{},{},{id:stone_slab,Damage:7,Count:1}]}
Entities, unlike blocks, can be moved freely, rather than only placed on a grid.