If you are looking for somewhere to train there are a few swimming clubs around

Check the Google search results

http://www.google.com.qa/search?q=swimming+clubs+doha&ie=utf-8&oe=utf-8&...

If it's just for fun then try in the Hotels all over town...almost all of them have a swimming pool. You will need to pay a fee but it's up to you.

Regards!!