It looks like you're looking for a colocation host. You basically rent out rack space in someone else's datacenter. Your options are mostly constrained by geographical location though so HN is probably not the best place to ask. Maybe try some IT professionals in your entourage?
I personally only have experience with Cologix and I can vouch for them. Good installations, good network speeds, etc. My previous employers' entire infrastructure runs there pretty much. They're based in North America.
However you don't necessarily need to access the rack yourself. I believe most will install the server for a fee if you ship it directly to them.
It's also pretty expensive. 2U of rack space can cost something like $99 USD a month. It's worth it on the long run if you need extremely beefy servers though.
We haven't tried those before, but largely looking for offering similar to EC2 reserved instance with the host swapped with a box we bought or leased.
The static public IP is the only hard requirement.