I have a flight to Istanbul via Saudi Airlines, and from there to Moldova via Air Moldova. The tickets are separate, and so my question is, since I don't have Turkish visa and I can't claim and recheck-in my luggage, is there a service to do that for me, in Istanbul Ataturk airport? I'm asking because there are such services in Dubai (eg:- Marhaba).
Luggage or not, you're unlikely to get on the plane to Istanbul.
Luggage is one aspect; the second one is that, if your connection goes awry for whatever reason, you're not "insured" and will be stuck in Istanbul. Unless you pay for a new onward ticket, Saudia would then have to return you to Saudi, a situation they want to avoid at all costs.
This is why most handling agents simply don't classify transit on separate bookings as transit.. In fact, though Marhaba offers a self-connection service in Dubai, I've observed people (who only had hand luggage to boot) being denied boarding on flights Munich/Zurich -> Dubai due to the aforementioned connection risks.
The only one who can give you a binding answer is the handling agent for Saudia at your departure airport, so you need to contact them, explain your situation and ask whether they'll check your bags through and let you board the flight.
There is no way on earth to my knowledge that allows non passengers to claim and transfer luggage between flights on different bookings, that’s like inviting sending not so nice devices and not having to be on the plane yourself, neither have I heard of any such segued offered by any third party. I might be mistaken but I am lazy and travelled the world for 30 years so if there is a lazy way I normally have an idea