Is it posible to fail TeamCity builds if artifacts are not found? I mean without writing any special script, i.e. does TeamCity has such built-in functionality?
No, you're going to have to implement that functionality into your build script.
i.e. if you are using MSBUILD .. after everything is complete run the following command:
<Error Condition="!Exists('someArtifact.txt')" Text="Error!!! Artifact doesn't exist" />