No, a private person in the United States cannot legally own a sea otter as a pet, because sea otters are protected by the federal Marine Mammal Protection Act of 1972.
The Act names sea otters in its definition of a marine mammal and places a moratorium on taking marine mammals and on importing them into the country. It is unlawful to take a marine mammal in U.S. waters or on U.S. land, to possess one that was taken illegally, or to buy, sell, or transport one for any purpose other than public display, scientific research, or enhancing the survival or recovery of the species. That is why the sea otters you see at aquariums are held under federal permits issued for display or research, not as private property. The U.S. Fish and Wildlife Service is the agency that administers sea otter protection under the Act. Breaking these rules can bring a civil penalty of up to $10,000 per violation, and a knowing violation can bring a criminal fine of up to $20,000, up to one year in prison, or both.
Source: NOAA Fisheries, Marine Mammal Protection Act (full text)
Image: Sea otter at Morro Bay, Mike Baird, CC BY 2.0, via Wikimedia Commons.