Running  your owb website, like you've described, is known as affiliate marketing.  You create a website, that is targetted at a particular subject/group of people, then you find advertisers who will pay to advertise on  your site (e.g. Google Adsense, or even individual companies depending on the website subject matter).

There are a lot of skills involved in doing this - all of which can be learnt free and online, but it isn't an overnight success story.  You have to make a start and do what you can do/understand today, then over time you improve what you have, you learn more and make changes.  Over time it can build up into a full-time income.

So, yes, it's possible, but you've got a large learning curve ahead of you.  If you want it, you can make it happen, but it is hard work at the start ... I do some of this and sometimes whole days are chewed up trying to overcome what seems like a small problem. 

There are a number of organisations now that will provide you with everything for free and provide you with everything  you need.  These can be a great way to start, for free, and build up your understanding of what it is, how it works, what works and what doesn't.

Here are some free ways to earn money online: http://www.real-work-at-home.co.uk/FREEWorkAtHomeSystems/tabid/123/language/en-GB/Default.aspx 

All of these will give you the experience/tools you need to be able to get started, both in terms of what is needed - and what you want to do.  Over time you'll probably find you end up doing 3-4 different things that all come together to bring in some cash.