Can you still have kids without ovaries?

no because then there wouldn't even be an egg for a sperm to fertilize.

AnswerYes, as long as you still have a uterus to implant the egg in. Ex-vivo fertilization or "artificial insemination" in which a doctor uses eggs from a donor, fertilizes the eggs using either your spouse's sperm or the sperm of another donor and implants the fertilized eggs back into your uterus, is a common procedure.

Also, if you still have one good ovary, it houses eggs that would be capable of fertilization from your partner and implantation into your uterus the old fashioned way. If both your ovaries are gone, however, the process above is your only option.