Running a search engine is very expensive though.

The federal government spends about $1.3B a year on advertising and another $37.5B on data collection, with Google being a major recipient of both budgets. Nationalization would save a small fortune.

And for the economic tailwinds that efficient Internet research provides, I'm willing to bet we'd see significant economic benefits that eclipse the base cost, not unlike with Amtrak or the USPS.

The Internet Archive is probably the closest thing we’ve got to something like this.

Them and Wikipedia, definitely. Both make for excellent models of non-profit free-at-point-of-use information services.