I've never bought anything on Amazon Kindle store and I don't have a Kindle. Is there any chance for me to buy a book there and convert it into a file format I could read on my non-kindle ebook reader which understands all kinds of DRM-free file formats?
Yes and no. If the e-book is DRM free, it is possible to convert it to other formats with programs like Calibre. However, if it does have DRM, then you would have to use software that would remove it. That software does exist, but I havent used it. FYI, Amazon's Kindle does not support the open e-book format called ePub. However, Calibre can convert ePub to Kindle friendly formats, as well as just about any other e-book format.
You could always download the Kindle reader application for Windows and use that instead of using the ebook reader you have.
Like @Keltari said Calibre won't work with ebooks that have DRM on them.
Does your non Kindle ebook reader support web browsing ?
If the answer is Yes (and you want kindle books), then I can recommend Kindle Cloud Reader (compatible with the Chrome & Firefox web browsers), no kindle device is required.
If the answer is no, can your ebook reader open PDFs ? If it can and you specifically want Kindle books, then I would recommend downloading the kindle reader application for your PC, and following an answer I posted under How to convert Kindle books into PDF format?
If your ebook reader cannot support pdf viewing then by far the simplest solution is the one recommended in Paul's answer.
Ars Technica IS your friend.
If you buy e-books from Amazon and want to engage in a bit of digital civil disobedience—by stripping the files’ DRM and making sure that Amazon can’t deny you access—we’re about to show you how. Yes, many parts of the Internet have known about this technique for some time now, but we feel that it bears mentioning again here.
But, before proceding, remember:
We will offer this caveat, however: it’s quite possible the technique we’re about to outline violates not only Amazon’s Terms of Service, but the Digital Millennium Copyright Act as well.
If you're still there, try following the steps on: DRM be damned: How to protect your Amazon e-books from being deleted.