"I would repeal all of Obamacare," Paul said. "The technical question though is whether or not - and I think this is why it's not an easy answer - the technical question is what would that mean? Can a state still have an exchange? We live in a 50 state union, so some states could have exchanges. They already did before Obamacare."
Democratic Kentucky Gov. Steve Beshear has defended the state-run exchange, which has been embraced as a success among residents in the state where the federal law is unpopular. Meanwhile, the state's other Republican U.S. Senator, Mitch McConnell, has said state officials could choose what to do with the exchange if the federal law were repealed.
Kentucky was among the states that chose to run its own exchange under the law, while other states left operation of the online marketplace up to the federal government. The marketplaces help people select insurance plans and determine if they are eligible for subsidies to lower their costs.
Under the health overhaul, states also had the choice of whether to expand Medicaid with the federal government paying most of the extra costs. Beshear chose to expand the joint federal and state health insurance program for the poor and disabled.
Republicans have seized on stories of the law's shortcomings, including some people who found out their doctors won't accept new plans issued through the marketplace.
Democratic candidates in November's midterm elections have found the subject tricky to discuss. In Kentucky, Democratic Senate candidate Alison Lundergan Grimes has pledged to fix the law, but would not say if she would have voted for it had she been in the Senate.
Her opponent, McConnell, has made repeal of the federal health care law a big part of his re-election campaign, but said last week that kynect was "unconnected" with efforts to repeal the Affordable Care Act. He later said it would be up to state officials to decide what to do with kynect.
Paul said much of the praise for Kentucky's health insurance exchange has been "because the website works," a reference to the glitch-plagued rollout of the federal health exchange website. Now that Kentucky has so many more people on Medicaid, Paul said he is concerned local hospitals won't have enough resources to treat them.
"I don't think we yet know how we'll handle the burden of the increased influx of those on Medicaid," he said.
Beshear has defended kynect, writing in an op-ed published on the Huffington Post that the program is working.
"Improved access to health care has had an immediate impact on the lives of these families. Statistics show that over 20,000 newly insured Kentuckians have sought care at hospitals since receiving coverage," Beshear wrote. "Over time, that care will help lift Kentucky off the bottom of nearly every health ranking."