The first thing that you will want to do is to get an old computer that you can wipe and then put linux on it. Failing that you can always get a Raspberry Pi, but for what you're spending to get a cute little manlet computer you could probably just buy some old person's computer at a yard sale and it will have way more horsepower.
In a pinch you can rent a private server but you'd be crazy to get something outside of Russia. Western hosting companies are too conservative for most of your needs. I've had decently luck with Hostinger. But in any of these cases you're just creating more work for yourself by not setting up your own server somewhere.
Your next step will be to turn your linux server into a DNS server. I found this tutorial online:
Seems legit. It's been a while since I have done this. You certainly will want to use some sort of upstream server unless you plan to email five hundred different domain admins to ask them what their IP addresses are, and then get them to ping you every time they change their address layout.
The upstream server that I have used in the past got sold out. While I haven't used Yandex for DNS before, I have used them profitably for email. They seem among the more stable of the Big Vodka tech companies. Here they have posted their tech settings.
Ok, so you have basic DNS set up. But so what? How is your DNS now any better than anyone else's DNS? Next you'll need to edit your zone records so that you can actually add the entries that your enemies are trying to cicrumvent. This is the fun part and it's easy:
My system shows 126.96.36.199 as the address for the Pirate Bay. So you would edit your zone record to add:
thepiratebay.org IN A 188.8.131.52
Technically you want to use tab characters, not spaces but I had trouble with the autoformatting.