You can get bare metal servers from a host on monthly or hourly terms.
You can get a 1/3rd rack cabinet at a colo facility and bring your own router, and use the colo's IPs
You can get a cage or a room at a colo and arrange for connecticity to your various upstreams and peers with your own ASN and IPs.
Only after all this is exhausted would I move towards and owned and operated datacenter where all the pieces are under company control and responsibility. (Although, if you rely on it, you're responsible for it, regardless of if you actually control it)