No that is not accurate according to politifact and it is the national archives that have control
Trump’s suggestion that Obama had personally kept millions of
documents including classified materials is wrong. The National
Archives and Records Administration, or NARA, said in response to
Trump’s statement that "former President Obama has no control over
where and how NARA stores the presidential records of his
The National Archives said it "assumed exclusive legal and physical
custody of Obama presidential records when President Barack Obama left
office in 2017."
The agency said about 30 million pages of unclassified records went to
a National Archives facility in the Chicago area. Classified Obama
presidential records are in a facility in the Washington, D.C., area.
The Obama administration turned over the documents to the national archive which was in charge of determining what to do with them.
The Obama Presidential Center in Chicago won’t be in the library
network operated by the National Archives, unlike other presidential
libraries. The center will be privately operated and built by the
Obama Foundation, which will raise money for the center. It remains
under construction with an expected opening date of 2025.
But the National Archives continue to own and control the documents.
During the digitization project, a memorandum of understanding between
the National Archives and the foundation said, "NARA will not be
transferring control, custody, or ownership over any of the Records to
the Foundation, the Vendor, or any other third party."
The ruling they give is that the statement is "pants on fire":
Trump said, "President Barack Hussein Obama kept 33 million pages of
documents, much of them classified. How many of them pertained to
nuclear? Word is, lots!"
Trump is wrong. News reports starting in 2016 showed that the National
Archives and Records Administration would oversee transfer of Obama’s
presidential records. The agency announced it would digitize the
records and that classified records were sent to a facility in College
Obama does not have them. We rate this statement Pants on Fire!
NARA Press Release August 12th 2022
The National Archives and Records Administration (NARA) assumed
exclusive legal and physical custody of Obama Presidential records
when President Barack Obama left office in 2017, in accordance with
the Presidential Records Act (PRA). NARA moved approximately 30
million pages of unclassified records to a NARA facility in the
Chicago area where they are maintained exclusively by NARA.
Additionally, NARA maintains the classified Obama Presidential records
in a NARA facility in the Washington, DC, area. As required by the
PRA, former President Obama has no control over where and how NARA
stores the Presidential records of his Administration.