It's very similar to spanish "cebolla" (you would pronounce it something like себо́ля). Or also the italian "cipolla" (чипо́лла).
Are Ukrainians very possessive about their onions?
My dictionary says цибуля is singular and цибулі is plural. (http://lcorp.ulif.org.ua/dictua/dictua.aspx). So, shouldn't this be translated as "Is this your onion ..." ?
Цибуля is uncountable. Countable variant is цибулина (~bulb), plural - цибулини. Цибуля can be sometimes used in the meaning of "цибулина", thus цибулі exists.