Is the Stack Exchange engine available?
Is it possible for me to make a Q&A site like Stack Overflow for a different topic? How much money it will cost? I have my own VPS; can I host the site there?
The Stack Exchange platform (for all intents and purposes) is not for sale.
If you'd like to make your own Q&A platform, you can try one of the open-source alternatives to Stack Exchange, or you can go through the Area51 proposal process and see if your idea has a community around it.
Look at Area51. It is a staging zone for new sites: