Thanks for the guide Yefim, that's good information!
I found this post because of the mention of the name Mackler. My family is also from the area around Bessarabia and Ukraine. I have a marriage record in the US showing my 2x great-grandfather was born in Odessa, but had no other information other than they came from "Polish Russia."
It took me years to find any records for them on Jewish Gen. First I had to find their Yiddish names, which I got from their headstone when I visiting their grave a couple of years ago. Then, even though I thought I was being inexact, I was still limiting my searches too much! I finally found the marriage record of my 3x great-grandparents in Kishinev. I had been doing phonetic and sounds like searches on Mackler and Mekler, but the spelling used by the Russian officials were Meykler and similar. Their first names (Favyesh and Frima) were spelled different enough that I wouldn't have found them without going slowly through the records and reading each one. Plus, I wasn't expecting to find them in Bessarabia and was too focused on Ukraine.
So now I have found records showing they married in Kishinev, they had my 2x great-grandfather in Odessa and birth records of other children, in Kishinev, showing they were registered in Medzhybizh. I've also found a DNA cousin and her 3x great-grandfather was born in Odessa and records indicate that he was my 3x great-grandfather's brother.
I just wish there were more records for my family, there just aren't that many available for these areas and I don't know what other places they might have been.
Terry, I'd love to get in touch with you, but I see no way to contact you outside this forum.