According to speisa.com, BBC News, and sources cited by this skeptics post, and many others, "rich Arab gulf states" accept none or almost none Syrian refugees:
The six wealthy Gulf countries - Qatar, the United Arab Emirates, Saudi Arabia, Kuwait and Bahrain - have offered to receive exactly zero Syrian refugees.
Or BBC News:
Without a visa, Syrians are not currently allowed to enter Arab countries except for Algeria, Mauritania, Sudan and Yemen.
However, other sources such as india.com and US News report that Saudi Arabia claims to have received millions of Syrians, but that they are statistically not considered refugees. This part of the claim is not addressed in the aforementioned Skeptics post:
Saudi Arabia announced on Friday it has received 2.5 million Syrians since the beginning of Syrian crisis in 2011, Saudi Press Agency reported late Friday.
...and U.S. News:
Saudi Arabia said it has taken in about 2.5 million Syrians on religious and humanitarian grounds in the years since the country's conflict began and has offered residency to hundreds of thousands, as it sought to rebut suggestions that oil-rich Gulf states should do more to address the plight of refugees fleeing civil war.
The official Saudi Press Agency quoted an unnamed official at the Foreign Ministry as saying the kingdom does not consider those it has taken in as refugees and does not house them in camps "in order to ensure their dignity and safety."
That's a rather large difference, between 0 and 2.5 million. And it seems like keeping track of 2.5 million people should be... possible.
Is there any reliable, objective source keeping track of this? Is it fair to say that the "rich Arab gulf states" have accepted none or almost no refugees from Syria? Is there objective evidence to suppport or shed doubt on the Saudi claim that Saudi Arabia has accepted many people from Syria?