Several layers to this:
- the feds dont care much. If your proposed airport conflicts with an airway or another airport, they label it as 'objectionable' on the chart but that is all they can pretty much do.
- some states require a license for any airport that goes beyond a 'temporary landing field' that is used for less than X days per year. The state may have obstacle clearance requirements and if you need to violate them, you need them to sign off on it
- your county may or may not require a zoning variance, special use permit or whatever it is called in your jurisdiction. It all depends on your counties zoning ordinance. Some may list the type of land where 'airport' is a permitted use, e.g. on light industrial. Others are silent on the issue and the only way to find out how they handle it is to either build it or to put in for a permit and see what they have to say.
- if you need to do any grading or changes to drainage, the local soil conservation district, drainage district etc. may have a say on what you can and cannot do.
- if your land is zoned agricultural, a landing field may be covered under your states 'right to farm' legislation. That is if you can make a case that it is part of an active farming/ranching operation. You need the field for the cropduster or to land the cub you use to look for lost cows ;-)
So, it all depends. County permitting is going to create 95% of the potential obstacles.