One of the things that seems to stand out of place in the Constitution with the rest of the document is the section compelling the government to operate a post office. Especially since there are several companies that are able to profitably compete doing post.
Presumably they chose to have a government post office because then all addresses in the country would be served, no exceptions, and on one uniform system. Furthermore, doing it this way makes the mail system as stable as possible, setting it high above potential business crashes since it will exist as long as the government does, fostering commerce by ensuring long-term stability and confidence.
The twist is that a post office sounds exactly like the main utility of the 1700's and the thing I've always thought that the Constitution was designed for a hands-off central government!
As far as things they could have also made public utilities, the only things I can think of is public fire departments which Ben Franklin first organized or city waters systems or perhaps canals, but FD's and city plumbing seem best left to local governments and canals left to states given the size of states relative to geography.
The post office article leads to post roads, which I didn't know about before and they seem to be the forerunners of the government building highways and interstates, which were officially built for national defense. Of course the military, the government itself, a patent system, and the justice/court system are public utilities.
If the government had been formed a century ago, would they have counted railroads or electricity or telephones as some sort of public utility? If nowadays, would they have explicitly included things like healthcare, education, cell phone networks, or internet access as public utilities, best handled nationally to various degrees? It seems like the same arguments for a post office could be made for any of those things.
Whether or not they intended the federal government to go into business in new fields, the Constitution they wrote is ambiguous enough to allow it to expand to encompass them over time. The government runs and regulates ports and airports, national parks, a space agency, and smart grids.
Despite what I've written, after skimming the history of the British Post Office, in comparison our government looks restrained. Their P.O. had a monopoly over all communication and they nationalized telegraph companies and ran the telephone system. Perhaps a difference in approach is related to how we've always been expanding into wilderness whereas in the same time frame the UK has had the same cities and it was figured that private business could best deploy resources to connect new areas.