Can you keep a blue and flathead catfish in Ohio?

Below 35 inches neither species carries a limit.

ODNR source page

One blue and one flathead a day at 35 inches or longer - the allowance is per species, and it is a statewide total rather than a per-water one.

ODNR source page

What we left out on purpose

• The Ohio River is deliberately left out of this page. Ohio shares it with Kentucky and West Virginia under two separate interstate agreements, and the rules differ between the western and eastern units - walleye carry a 14-inch minimum in the western unit and none in the eastern one, on facing pages of the same booklet. Those limits are real but the question of which agency's rules apply to which angler is not something we were able to verify cleanly, so we publish nothing rather than half of it.
• Crappie, bluegill and northern pike appear nowhere in Ohio's statewide table. That is not the same as saying they are unlimited - ODNR simply lists no statewide size or bag limit for them, while setting limits on individual named waters. Check the water-specific list for the lake you are fishing.
• The muskellunge entry says no statewide minimum length because that is what the regulations booklet states, in both its English and Spanish editions, and Ohio's Administrative Code carries no statewide muskellunge length rule either. Many other sources quote 30 inches statewide; that figure belongs to the Ohio River and Pymatuning Lake.
• Walleye and yellow perch limits for Lake Erie come from a separate ODNR sheet rather than the main booklet, which defers them with the words 'to be announced late April'. ODNR resets these numbers every spring, so they are the first thing to re-check each April.