Are GA airports typically accessible 24/7?
The general answer is that the airport manager is in charge of ground access (unless the TSA is involved; see below) and individual FBOs can open or not whenever they like. There are no standard federal rules for the same reason that there aren't any for when gas stations should open and close: they're private (or state-owned) businesses.

So before going somewhere new it's always good to call and ask about opening hours, access etc. If there are multiple FBOs at the field, you can call them all: one might be open 24 hours a day even if another one isn't.

One specific point about Salina is that the TSA may be limiting access. The AOPA airport directory says that KSLN is under security directive SD-8G, i.e. the TSA limits access to the apron. That means you might need an authorized escort - most likely someone from the FBO - to and from your aircraft. If the FBO shuts at night there may be no one there, but FBOs in general are usually good at making special arrangements if needed.

And although your question seems to be about ground access, I guess it's worth stating the obvious: airports are accessible 24x7 from the air. Nothing stops you landing at 3am, filling up at the self-service pumps and then taking off again. But watch out for local practices like voluntary noise abatement curfews, e.g. at KPDK:


  Pilots are encouraged not to fly between the hours of 11 p.m. and 6
  a.m. Due to FAA Regulations, PDK cannot implement a mandatory curfew
  at this time.


It's still legal to land or depart there during curfew hours, but you'll really annoy the airport management, FBOs, and local pilots who have to deal with the complaints from local residents.