Another option, and the one I just started using, is a mini-pc. More power than a Raspberry Pi based system, at a similar price point. I installed Proxmox and then HAOS as a VM. Very easy to do.
I have HA running on a RPI4 for a few years now. I do use an external USB ssd for its storage, the ssd cards arent great for that constant log book writing. I would avoid running it on a Synology; if your Synology fails (though they are quite reliable, they do fail), you don’t want to have to explain to your significant other that the NAS took down the house.
Most my house lighting is on HA, two sprinkler systems, HVAC, incoming, outgoing energy and solar production, a bunch of sensors and a few door locks and some other odds and ends… all on that humble rpi4.
Take into account the hardware dongle’s you might need or want to use.
Z-Wave, Zigbee, Thread, etc. typically are plug-in using USB. Virtual hosted HA instances are more complicated when dealing with pass-through access to physical USB ports versus a dedicated computer box (mini PC, rPI, older MAC or PC, etc.)
My advice - if you haven’t used HA at all yet, I would load a test version on something you already have that is free. Try it out with existing Wi-Fi or Ethernet connected devices first, so no USB/dongle issues.
If you like HA, then move on to a “production install” and choose the appropriate host/platform at that point taking into consideration immediate use and future growth.
I would recommend running the Home Assistant OS on either a Pi or a VM, at least to start. The OS option is their preferred option and it gets some features, such as add-ons and easier backups, that aren’t available if you just run HA in Docker. If your Synology supports VMs, I think running HA OS in a VM on it is also fine for starting out. You can always move to new hardware if you outgrow whatever you initially run it on.
Personally, I run HA OS in a VM on a mini-PC. Besides using HA’s built-in backups, I have snapshots of the VM taken every night. This way if something gets really messed up, I can always restore the entire thing in one shot.
Get a mini pc. Something like a beelink s12 pro (n100 or n150 chip) is easy to run it bare metal. Or for same price as home assistant device throw Proxmox on it, use proxmox helper scripts and you can run home assistant in a VM (and a ton of other services all at once), and easily back up the VM or migrate to new hardware in the future. It’s best of all worlds.
There is a big contiuum of methods to run HA (or other appliance-like tools such as PieHole and others).
Just a reminder that there is a huge spectrum of user expertise, budget, and desire that makes Dockers, VM, ProxMox, homelab server PCs, etc. everything from the greatest-thing-since-sliced-bread to an absolute Rube Goldberg nightmare depending upon one’s personal desire and skills.
As an FYI, a spare/older Mac or Mac Mini can also be a host for Dockers and VMs if you have the hardware lying around. You can even load Linux directly and go from there if one is so inclined.
Personally, not just for client work but for my own tinkering, I prefer physical little boxes/processors dedicated to one function each.
I still chuckle at the horror stories of expert hobbyist/hackers with skills way beyond my abilities posting the inevitable “just spent the whole weekend rebuilding my server” or “My big-ass NAS/Docker/VM server went down and my entire home network and stuff was SOL until I finally get it fixed”.
I have mine on a mini PC - purely because at the time, they were easier to get hold of compared to Pi’s. I started with it ion Docker on my Synology, but quickly moved to it’s own hardware when I bought myself a Zigbee dongle to use with it.
When I got mine, I moved everything to it - but I’ve since moved the Hue back to their own Zigbee network and hub, as I experienced an issue with the server a few months back and all the lighting in the house stopped working. Keeping it on the Hue hub allows me to still control it using Home Assistant but I can also remain in control of it if the HA server shuts down for whatever reason.