I've written a simple Angular 2 app and published it on a hosting provider's Apache server. Is it possible that my site will be crawled by search engines? I googled that the site needs to be rendered server side, but I haven't been able to find out anything else about it.
The GoogleBot can crawl your website just fine, since it also executes javascript running on that site.
In the case of Angular2 it is clear that the GoogleBot's JavaScript capabilities are good enough to render the rather sophisticated front end framework. The demo application rendering shows that GoogleBot does render everything ok - even without Universal Rendering.
Using the Google Webmaster tools you can get the GoogleBot to visit your page and render your site.