After i've eaten some home made pickled onions (made with half water and
half Sarsons malt vinegar) I'm left with the vinegary and 'cloudy' juice.

I thought I would like to use this juice to mix with olive oil and raw
garlic to make a salad dressing. (Vinegrette).

Is there an easy way to *filter* the particles out of the juice that make it
cloudy?

Maybe I could use a white paper kitchen towel. But could it be that I would
be washing some chemicals out of the towel that would not be good to eat?
Any suggestions on how to filter these very fine particles appreciated.
Thanks.