I checked the docs better, as suggested by Ville Säävuori, and I decided to add the strikethrough like this:

.. role:: strike
    :class: strike

In the document, this can be applied as follows:

:strike:`This text is crossed out`

Then in my css file I have an entry:

.strike {
  text-decoration: line-through;
}

There is at least three ways of doing it:

.. role:: strike

An example of :strike:`strike through text`.

.. container:: strike

   Here the full block of test is striked through.

An undecorated paragraph.

.. class:: strike

This paragraph too is is striked through.

.. admonition:: cancelled
   :class: strike

I strike through cancelled text.

After applying rst2html you get:

<p>An example of <span class="strike">strike through text</span>.</p>
<div class="strike container">
Here the full block of test is striked through.</div>
<p>An undecorated paragraph.</p>
<p class="strike">This paragraph too is is striked through.</p>
<div class="strike admonition">
<p class="first admonition-title">cancelled</p>
<p class="last">I strike through cancelled text.</p>

You use them with a style

.strike {
  text-decoration: line-through;
}

Here I have taken the admonition directive as example but any directive that allow the :class: option would do.

As it generates a span the role directive is the only one that allow to apply your style to a part of a paragraph.

It is redundant to add a class strike to a directive also named strike, as suggest Gozzilli, because the directive name is the default class for the html output.

I have checked these syntax both with rest2html and Sphinx. But while everything works as expected with rest2html the class directive fail with Sphinx. You have to replace it with

.. rst-class:: strike

This paragraph too is is striked through.

This is only stated in a small footnote of Sphinx reSt Primer.