Yes, this is possible if your friend can manage to get a letter from his sponsor. His sponsor needs to issue a letter with your name and passport number saying that he has no objection you guys to visit saudi arabia for a business trip, once your friend get this letter, he needs to get it stamped from Chamber of Commerce in Riyadh and then let him email or fax it to the same to you (no need for original), then you apply a visa in Saudi Embassy here in Qatar along with your passport, return ticket back to Qatar with nominal fee. Thats all you will get the entry permit for a month or so. But I don't think this will work if childredn are accompanying you.