Unfortunately, "middle" sized isn't very descriptive.
Based on what requirements you choose (server speed, RAM, traffic, etc.) the prices will very. From a very small , simple server, you are looking at probably about $250-1200 per month.
You should investigate more what you need, Your server requirements will change when you incorporate software you want to run, hardware you prefer, etc. There are some good sites, just google them and you can find. Try googling "Hosted Server" or "Hosted Server Costs".
Hope this helps!
It also depends on location of server. For instance the cost of web hosting in Wales
can be different to the cost of Bristol hosting
Except for the technical specifications, the service levels (e.g. availability, support, etc.) will have a large impact on the price. Just an simplified example.
Say the server cost $3600 to buy and they use a writeoff period of 3 years. the cost per month for hardware is 100$ a month (not taking in account interest, etc.). The rest of the cost are software (e.g. OS, but also the mgt tools), maintenance and mostly management cost. Simple management of a server may be 1 or 2 hours a month (just for monitoring), more extensive management is usually double that (3-4 hours a month). And higher service levels may require extra investments (e.g. extra hardware on a different site in case one DC site goes down.
Another aspect is if you want to make use of a physical or a logical (virtualized) server. This again also depends on how do you define hosting. Is it just the floor, racks, power and hardware. Or is it also the OS and virtualization layer. In other words. From something that is close to housing or to a platform where you can run your applications on.
I agree with the range above ($250-1200) a month for an Intel server (Windows, Linux). Other platforms are usually (a lot) more expensive.