VueJs page gives 404 on refresh

Assuming your entry point is the index method of your HomeController, you just need to add this route at the bottom of your routes/web.php file.

Route::get('{any}', function () {
    return view('homepage');
})->where('any','.*');

This isn't an issue with Vue.

When I visit the webpage http://blog.sidneylab.com/post/quasi-non-est-magnam-quae-aut-omnis-nisi (the first post on your website), I get a 404 error from the server itself. Same with anything else on /post/

That means your route handler isn't sending the relevant Vue index or code when a route matches. Ensure it does this with either the /post/ URL or just add a catch-all handler at the bottom of your default route handler:

Route::get("{any}", "DefaultHandler@myMethod")->where("any", ".*");