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Investing in a server with mass storage would "pay for itself" in less than a year, compared to what I'm currently renting (I'm low key scared to look up the prices of DDR5 RAM and NVMe drives though). Since I plan to maintain TankieTube "forever", it seems like the best option.
I'm so ready to ditch BackBlaze because their timeout errors are causing ~90% of the current problems with the website (external storage move failures and buffering problems). 
I have plenty of experience assembling computers and the thought of building a server is really fun, but I've never used colocation before.
Questions/Thoughts/Concerns:
- Do datacenters let you walk inside to maintain your own server? There is a datacenter in my home city, which would be convenient, but using it would effectively soft-doxx my location. Right now "Burgerland" is as specific as I publicly reveal.
- If I ship the server to a more remote location, how would I replace failed drives? Is that a commonly provided service? Would using a datacenter within ~2 hours driving distance be the best compromise between accessibility and location obfuscation?
- Is paying with Monero an option? Is it a good idea? Could I mail replacement drives directly to the datacenter without revealing my home return address?
It looks like I'll need NVMe drives in something called the U.2 form factor (instead of M.2) in order to enable hot swapping. TIL.
First few questions have been answered adequately already, and I can't answer the third, but regarding uptime vs. distance; consider using more than one machine. Depending on your budget it might be worth it to have a couple or more lower-spec machines behind a load balancing / failover proxy. That would give you a fair bit of leeway to get out to the location while maintaining uptime. Also you can use one to troubleshoot the other remotely.
Are you talking about multiple servers in the same datacenter, or space across the globe?
Same datacenter. I don't expect tankietube to scale to a global CDN quite yet.
How do I keep the PostgreSQL databases in sync?
I don't think PeerTube is made to run more than one of the same instance? If it were to work, for a low amount of servers you could maybe just connect them all to one database. But for more than one PostgreSQL server, this page might be an introduction:^[1]^ https://www.postgresql.org/docs/current/high-availability.html
1: I do not know about this, I just know it is possible and looked for it
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I don't think so either. It has a native redundancy feature in which different instances mirror videos to distribute the demand.
https://docs.joinpeertube.org/admin/following-instances
I've thought about creating a "dummy" instance with user registration disabled which exists solely to mirror TankieTube videos.
If what you want is to distribute the video delivery demand, then a CDN is more what you're looking for. Setting up another instance would be to help with everything else peertube does, showing lists of vids, channels, showing the vid page (but not the vid delivery), all of that stuff, and uploads (but of course not transcoding).
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I've never done it with postgres, and it's been a while since I've done this kind of thing in general. I'm also not familiar with your deployment, but postgres has options for all kinds of failover and load balancing scenarios.