I want to make a base for my NFG, preferably in the Core, but I'm not sure if I need permission from the government. I don't even want a system, just some open space. Can/should I do it?
If you are talking about a city. Then you need permission to slab from who ever controls the planet. If you are wanting to build on a city already slabbed you need permission from the city owner. If you are wanting a station you need permission from who ever controls the system, unless you are building a station in deep space then you need no ones permission.
No their are no restrictions currently for building in deep space. so unlike Faction controlled systems you will not have to send a Station Permission request or such.
However any unapproved stations in sectors are likely to be used for target practice come combat. If you have any doubt if a government would allow your station then it may be advisable to ask first.