Government owned businesses now?
No
There are a lot of government owned businesses that "compete" with private businesses.
Public schools compete with private schools.
Public parks and gardens compete with privately owned parks and gardens that charge an admission fee. Heck - some of the public ones charge an admission fee, but receive government funding.
Publicly owned museums.
Municipal utilities.
The United States Postal Service competes with private delivery companies.
Amtrak and other publicly owned passenger rail.
Airports.
Public transit.
Lotteries compete with private gambling.
Public ports.
I'm pretty sure that's not an exhaustive list.