Is there any precedent for that happening?
I know California tried to do that during the Dust Bowl to keep certain “undesirables” and “Okies” out. Legally, I think only the US congress can close state borders re: the Commerce Clause in the US Constitution:
I am doubtful most states would have the resources to do it. I mean, you could certainly close off Hawaii or Alaska, more or less, but the continguous states mostly are so open to other states that no state will have the manpower to do it. Scan around the borders of a place like Pennsylvania, Texas or Indiana. There’s just a limitless number of roads in and out.
Do-re-mi - WW Guthrie (from memory)
Lots of folks back East, they say, are leavin’ home every day,
Hittin’ the hot old dusty way to the California line
Across the desert sand they roll, headin’ out of that old Dust Bowl
They think they’re goin’ to a Sugar Bowl but here’s what they find
Well, the po-lice at the port of entry say
"You’re number 14,000 for today.
“If you ain’t got that dough-ray-me, boys, if you ain’t got that do-re-mi
Well, you better go back to beautiful Texas, Alabama Kansas Georgia, Tennessee
California is a Garden of Eden, a paradise to live in or see
But believe it or not, you won’t find it so hot
If you ain’t got that dough-ray-me”
Los Angeles sent deputy sheriffs to the state line to intercept migrants. Price of admission: all you’ve got. My Okie step-family had barely enough. Even if states close borders, those with enough do-re-mi will get in or out. Money wins.
Don’t know about a state, but we do have an island here in Maine that closed themselves to outsiders. Judging by your username/location, you probably heard about that, and maybe it inspired the thread.
It doesn’t mean that they have to man every crossing, just block the road, jersey barriers, trucks, dirt piles. It can be done.
I also recall reading about the Alaska Gold Rush they would keep people out unless they could prove they had enough money and/or supplies to last them a certain amount of time. But Alaska was a territory back then. And it isn’t quite the same thing.
Mrs. J. and I were just speculating on whether, if Ohio shut down its borders, more people would be trying to get in or out. 
In Sinclair Lewis’ “It Can’t Happen Here” the main protagonist tries to flee his home state by sneaking down a back road to the border in the middle of the night, but is thwarted by an unexpected checkpoint.
There are lots of “non-essential” government employees and National Guard types with time on their hands, so theoretically they could patrol all the roads leading out/in. With cases in all 50 states already, I don’t see it happening.
That was Canada that had that requirement. The Klondike, where that gold rush was, is a steam in the Yukon Territory. And the requirement was a year’s supply of food. Nothing about money.
The outer banks in NC routinely restrict non-residents from access during and after a state of emergency. Normally it’s for hurricanes, but they’re using the same process and laws for the current crisis: https://wset.com/news/coronavirus/visitors-banned-from-outer-banks-amid-coronavirus-outbreak
I don’t think that there is any constitutional issue with restricting access to an area for non-residents during an emergency, but it’s wildly impractical for any but the tiniest states plus Alaska and Hawaii. There’s the basic issue of manpower to cover borders and the difficulty of setting up entry permits and exceptions since you don’t want to stop trucks from delivering shipments. But also a lot of state borders have local areas that don’t stop their ‘localness’ at the border, and trying to stop people from wandering back and forth like they usually do can be difficult.
Look at Pennsylvania on a map. Even if you blocked every last road into PA, how would you block hikers from walking in?
Limiting travel seems to be effective, but I don’t understand the fixation on borders (whether it’s international or inter-state). The virus is already spreading in every US state.
This. It’s already here. Sure, many individual cities are so far untouched, but barring off cities would be even more insane than barring off states. Way too many people are still working, and need to cross city/state lines.
I live at a point where the corners of 3 states come together, and it’s all considered one big metro area. I live in one state, work in another, and frequently travel to the 3rd (hey, it’s where the closest Walmart is). As long as I’m continuing to limit my exposure to as few people as possible, as we’re all being encouraged to do, no one is being put at any additional risk.