As an EU citizen, I have the right to reunite with my family members, even if they are non-EU citizens. So if an embassy isn't fulfilling their responsibilities and never gives a valid reason for not considering my case, or even respond to my emails and phone calls, can I sue them?
You cannot "sue" the embassy per se and you certainly cannot (effectively) do that in front of foreign courts. What you can do will depend entirely on the country in question (even within the EU) but basically it will be similar to what you can do against adverse decisions (or silence) from the authorities back in the sending country (not the country where the embassy is located but the country it represents).
The details vary a lot, some countries, including Italy, have an entirely separate court system (administrative courts) for that purpose. But if it goes that far, you will definitely need a lawyer and he or she should be able to advise you on the options available (going through some informal or formal appeal process might also be worthwhile or even necessary before going to court).
I believe you're asking the wrong question. Assuming your main goal is to actually get your family EU citizenship/residence/entrance, what you may need to consider is legal action within the legal system of the specific EU state of which you are a resident (and which you also want your family to move to). That is, if an administrative organ is not carrying out its function, you can usually ask a court of law to order that an action be taken.
Of course, you would need to consult a lawyer or at least someone with relevant legal experience on this matter.
If you want to try to "punish" or demand some sort of accountability or compensation, that may or may not be possible, but as other answers suggest it will definitely be very difficult. Try getting in contact with civil society legal-oriented NGOs which deal with immigration rights, which might be able to suggest avenues of struggle against such practices.
Finally, remember that you are obviously biased in favor of yourself, so before you consider any 'offensive' action (such as a lawsuit) you should be able to convince neutral parties well enough of the validity of your arguments against how the embassy handled your case.