Most of the servers you see running 24 hours a day are rented from companies that host game servers, such as
Some people pay every month so other people like you can play for free on a professionally-hosted server. Isn't that nice of us?
You can run a server 24/7 from home if you have a fast cable modem or DSL Internet connection and a spare computer that can run the dedicated server program. But if you try playing on the same computer that is running the server, or if you try doing other things at the same time on the same computer, or if someone is surfing the Web or uploading\downloading files, it will cause lag.
You can get the dedicated server programs for Windows or Linux (make sure you get the same one that matches your operating system) here