[nginx] How to configure nginx to enable kinda 'file browser' mode?

You should try HttpAutoindexModule.

Set autoindex option to on. It is off by default.

Your example configuration should be ok

location /{ 
   root /home/yozloy/html/; 
   index index.html; 
   autoindex on;
}

Without autoindex option you should be getting Error 403 for requests that end with / on directories that do not have an index.html file. With this option you should be getting a simple listing:

<html>
<head><title>Index of /</title></head>
<body bgcolor="white">
<h1>Index of /test/</h1><hr><pre><a href="../">../</a>
<a href="test.txt">test.txt</a>                 19-May-2012 10:43            0
</pre><hr></body>
</html>

Edit: Updated the listing to delete any references to test