It took me some time to install Home Assistant on my Synology NAS. I was playing around with Docker, but it did not give me the satisfied installation of Home Assistant I was looking for. So I ended up using Home Assistant Core.
Login to your Synology NAS and go to the Package Center. Go to settings and see if SynoCummunity.com is allready configured. If not, add it to your Package Center.
After the configuration, go to the Community and search for Home Assistant Core, and hit Install.
It will ask you to also install Phyton 3.8. Just do it.