I'd not necessarily support 100% government control, but I feel there is a compelling interest for the government as an optional provider in the following markets (due to the presence of monopoly conditions or potential for coercive market power, dissimilar market access or where market failure would be catastrophic):
Education (primary through tertiary)
Healthcare and welfare
Utilities
Local transportation
Emergency services
Insurance
Postal service
Government options need not necessarily be public sector style options.
One of the better models is how Canada manages its air traffic control services, via a private corporation run on a not-for-profit basis via a public charter that requires a act of parliament to change.
There's also 'government as shareholder' models, where the provider is partially privatised but the government retains either a majority shareholding or a majority voting share (or a golden/blocking share).