Instead of making up something about an illness, just always politely say no when offered any alcohol. It is your business what you eat and drink. You do not have to make excuses for your likes and dislikes. Especially at 21 you are an adult and know your preferences. You may want to avoid being in any situation where alcohol is being served, but in any case, be consistent with your beliefs and your answer.
If you are a female say that alcohol reacts badly with the birth control pills you are taking and if you are male tell them you have been drinking protein powder shakes or taking multi vitamins and the label says that multi vitamins or protein can have a bad side effect with alchohol and can make you sick or you can just be blunt and say you're not interested nobody can force you do something you don't want if you don't want to drink who cares all you have to say is I'm practising sobriety and I really do not want to drink any alchohol not interested no thank you and that's all really ...
Last time I got real psychological pressure from a friend to drink it turned out he was an alcoholic and was just super-good at laying on the guilt and pressure to get everyone to drink with him. I mean, I usually don't care what people think, but this guy had me feeling guilty and embarrassed over turning hm down! If you are feeling like you need more than just a simple "No" to get out from under the relatives' insistence, it might be because they are alcoholics and that is why they press you all the time.
If so, you address the issue directly. "I've seen so many bad things come from drinking, I'm not going to." If there is acknowledged alcoholism in the family that has caused problems, they could probably understand that.
If you think being that upfront could cause a big blowup, rather than saying you have a medical condition that prevents you drinking, you could instead say you're on a drug that interacts badly with alcohol and aren't drinking. If they demand to know what one, say Antabuse. If they know what it is, that will shut them up. (It's a drug that makes you feel really awful if you drink alcohol, taken by alcoholics so they will stop drinking.)
I would say to just speak up and tell them that you prefer not to be pressured about your choice to abstain from drinking. Bring along something to drink, like, tonic water and orange juice, and mix yourself a drink of it, and enjoy yourself.