My business is… troublesome.
What I do is legal in every country on earth, but some people don't like it, and make it so tough on my poor ISPs that I am forced to go looking for new providers more often than I would like.
The only option I know of for somebody in my position is called "bulletproof hosting" (think wikileaks) and it isn't cheap.
The only thing that makes bulletproof hosting expensive, or anything different than typical hosting, is their legal stance when responding to "abuse" complaints via email. Your typical host will get fed-up after 5 or 10 no matter what the reason, and a bulletproof host will take the time to look through the legality of the matter and make a decision based on that.
As far as I know, these abuse emails are directly tied to the ip address on which their server sits, because they "own" that ip address and have the ability to lease it out to me on my expensive bulletproof server.
If I could answer them personally, I would save another company the trouble and hopefully save myself some money along the way.
How can I become the bulletproof host? Just rent out a room in my local DC and ask where to get the IPs from?
P.S. No… just because I know this will be the first question everyone asks – I am not some spammer or rule 34 pornographer, what I do is legal in every country. I said "think wikileaks"!
EDIT: Thank you once again for all of the amazing responses. Don't know why I've been lighting fires here recently. Thanks to everyone who saw through the smoke and provided me with meaningful answers.
Best Answer
You need to apply and be granted your own IP allocation by your local registry like RIPE or APNIC.
They require annual fees, and you need to justify your requirement (yours is legit). They will assign you an Autonomous System number and a range of IP addresses.
You must then find people to peer with (in a datacenter usually), preferably more than one. You then publish your BGP routes using your AS provided with your IPs via your new peer links.
You also should allocate a DNS server to provide PTR records for your allocation.
None of it is cheap, it requires expensive subscriptions, expensive peer links, expensive hardware (routers to do the BGP peering) and a fair amount of networking knowledge.
What I suggest is that you hire a contractor to set up the initial network and peering and firewall, etc to take the pressure off you having to learn and maintain it all, so you can concentrate on... Whatever it is that you host.