The short answer is no, it won't cost you anything.

Depending on your needs, the cost and how you spend it can affect the final price. If you want the server to run close to 24/7 and you dont want to use your pc there are generally two options available to you.

You can rent a server on a server host, depending on your needs this could range from a few bucks per month to 10 to 15 bucks per month. This is the cheapest, but it will be more expensive in the end.

You buy a small laptop, such as a raspberry pi with extra hardware modules, and you set it up yourself. You will need to set it up manually and cost around 80-100 dollars. After that, there should be no additional cost. Sometimes (Hardware may break, but let's not worry about that for now.

If you want to continue using it from your PC, you can purchase a wired connection. This will stabilize the connection and reduce lag. This will add cost to your computer and take some time to set up.