Homesteading is the act of claiming unowned land, and, by transformation of the land (building, farming, enclosing, etc.) claiming the land as first user (or first user after abandonment) and thus becoming the owner of the land. In a free society, former "public" property would all be considered unowned, as would some property obtained through force or state privilege, and could be homesteaded.
In a free society, people that use infrastructure will homestead and maintain it (and perhaps charge for use; fair deal). If nobody cares enough to do so, then the infrastructure probably shouldn't have been built in the first place (some places have six-lane highways for no rational reason, just politics; or bridges to nowhere).
If 
the state vanished, what about existing claims to property?
Regarding respecting property: I agree with Rothbard's idea () of respecting it unless you can show it was taken or acquired wrongly or that there is someone with a higher claim (sometimes those are the same thing). Examples:
- Property taken from someone by the state (eminent domain), and the owner is still alive: they should get their property back (but pay back whatever the state gave them for it, or destroy the money).
Property stolen by the state and sold to someone, but there are no living heirs—current owner keeps it, _unless_ they had a part in the state stealing it, then I would expect arbitrators not to recognize the thief's claim and treat the land as unowned and homsteadable.
Q: The situation I have in mind is someone purchasing, say, 1000 acres of land from the government and just sitting on it, and then "when anarchy happens" or whatever do they have the "higher claim" than the people who actually homestead it?
A: That depends how the government got it—if it was unowned, and the purchaser could have (and would have) instead just claimed the unowned land and used it, then it would work out the same. If stolen, then it should go back to the owner, and if the money exists, the new purchaser should get it back from the former state (even if they are unable to do so, they do not have a better claim on the land than the victim of state theft; they are sadly out of luck for buying from a thief; and given that it's hard to hide appropriation of land today, hardly innocent victims; they would know who the state had taken it from).
But you said "just sitting on it" vs. "homesteaders". The homesteaders would seem to have a higher claim if the purchaser isn't using it; but I'm worried about the slippery slope there, not being a fan of restricting ownership to "occupancy and use".
However, in the case of the person buying 1000 acres, the state was obviously selling it for some price perhaps out of reach of the homesteaders, who would have homesteaded the land first if not for that artificial (and violence-backed) price barrier. Thus their claim seems more legitimate, because they would have gotten there first and used the land if not for the state.
I.e., the person buying from the state is making use of the state's arbitrary force-backed claim. If they had instead just been there first and homesteaded the land, clearly posting their claim, that would be different. But that wasn't the example, nor do I think it would be common: in all cases of unowned land, there will be competition for it and people will have to either divide up the land or share ownership. It's never going to be, "La, la, look, I wandered into 1000 acres of arable land and nobody else around, I guess it's all mine" <puts up signs>—unless the person had to explore and take risks to get to the land, in which case they've labored for their homesteading advantage.
See also: Rothbard's For A New Liberty, "Property and Exchange", pp.30-37; his The Ethics of Liberty, chapter 9, "Property and Criminality" (especially the example of "Ruritania" and the "privatized state" and concepts of unjust and just land ownership), chapter 10, "The Problem of Land Theft", and chapter 11, "Land Monopoly, Past and Present".