Took me too long to find docs, so a link:
https://developers.home-assistant.io/docs/add-ons/tutorial/

And https://developers.home-assistant.io/docs/add-ons/configuration#add-on-config contains:

privileged ... Privilege for access to hardware/system. Available access: NET_ADMIN, SYS_ADMIN, SYS_RAWIO, SYS_TIME, SYS_NICE, SYS_RESOURCE, SYS_PTRACE, SYS_MODULE or DAC_READ_SEARCH

and

full_access .... Give full access to hardware like the privileged mode in Docker. Works only for not protected add-ons.

Between the two of them, should be able to do all the right tun stuff.

https://github.com/tsujamin/hass-addons worked for me, including the unmerged PR to bump Tailscale to 1.4.4

Thanks! I hadn't found that add-on. I've installed 1.2.10 using https://github.com/tsujamin/hass-addons
It seems like it is working. I'll try forking the repository and applying the 1.4.4 PR next.

Update: My rPi froze again, even with Tailscale disabled. So it probably has to do with an earlier Home Assistant update I did on the same day as installing Tailscale.

I used https://github.com/ClayKaiser/hass-addons for a few weeks, and while the setup was a breeze, I am experiencing stability issues on my rPi running Home Assistant OS.

Without any known reason, the rPi just stops working. No automations run, no remotes work anymore. It does respond to ping, but the web interface, and SSH server (both on port 22 and 22222) don't accept any connections.

I need to power cycle to get it to work again. This setup ran for years without issues, so it's pretty weird. I did update Home Assistant Core and OS, before installing Tailscale, so maybe that's the cause.

For now I disabled Tailscale to see if it runs stable for a week. Reporting this here for people to find with similar issues.

@mlangenberg Do you use their supervisor setup? I had my dad install tailscale via deb, and then later found that the Pi's configuration (due to supervisor?) was causing dhcpcd to trash, aggressively re-leasing many times per second. Eventually the Pi just falls over. I'm not sure what exactly is the cause of the thrashing. But now I'm mentioning it since it sounds like you might have a similar issue. (journalctl --follow was enough to see it)

No it's actually home-assistant/core#47928, so totally unrelated.

Is someone actively working on this? If not, I might be able to help out.