Flying within 5 mi. of an airport IS possible. For anyone who has seen the movie
Top Gun, you have all seen what used to be Naval Air Station Miramar, and has since been transferred over to the Marines, now known as MCAS Miramar.
I bring this location up because there is a club called the Miramar Flyers that are literally across the freeway from MCAS Miramar.
If you look at the shot above (and you can see this yourself on Google Maps if you want to look it up in San Diego), the arrow points right to their runway. How do they get away with this? Lots and lots and LOTS of provisions. The military wants guys to know how to fly unmanned aircraft; why not have a location where they can fly on base? It IS a public club, but because it's on base, it's also a pain in the butt to get to fly there because you have to be escorted on base by someone who's in the military/has a sticker authorizing them to come on base.
So, do NOT be discouraged about being within a 5 mi. radius of an airport. Here's the key to it.
1) Find a spot, such as a park, where there are large, open areas to fly at, perhaps a soccer field or a baseball field that is not being used. I have a baseball field that is literally across the street from where I live, where I will occasionally fly drones. The catch with this is that you want to fly when there's nobody else there. I say this for safety, because you can't crash a plane into people that aren't there.

Also, people don't complain if they don't see you.
2) Look around at the site you've chosen, BEFORE you fly. Do you see lots of low flying aircraft? Are they out cruising around your potential flying area, flying low? If so, look elsewhere. It's not worth fighting over the airspace; the big planes ALWAYS have right of way, and it's just not safe.
3) How high are you planning on flying? I fly with a group of drone flyers at Kit Carson Park in Escondido, every other weekend. It's somewhat close to some local municipal airports, but one of the big things for us with flying there is that there are a BUNCH of trees surrounding the field we fly over. Flying objects and ground objects (such as 40' eucalyptus trees) do not mix. If you keep your flights below the treetops, usually you won't have a problem.
4) Do you still need to get permission to fly in one of these areas, if it is near an airport? According to the FAA, YES. You are supposed to contact the tower and make them aware of the flights you are going to take. Are there exceptions? YES. Two of the fields I fly at have airspace notifications for pilots. One of the areas where we fly is in a county park, and it is at the start of a box canyon about 2 mi from the local municipal airport. Most pilots who fly over the field at that point are flying 1000' or higher in order to avoid colliding with the hills and/or turbulence coming off of the hills in that area.
I will be honest that I truly think you are better off flying with a club. Almost every club I've come across has had trainers to teach people how to fly. They'll help you buddy box and learn to fly your plane, hopefully without crashing it (sometimes, things just go badly, like wind that comes up and blows your plane into a tree, or a servo wire gets hot and fails, or that little glue joint for the control horn of the elevator lets go during the flight, and down comes the plane). The clubs have events (at least, mine does - we do night flys, camping at the field, combat, altitude games, shuffleboard with combat wings, BBQs, and more) that really kind of pay off. And yes, they DO require AMA membership; they do that for insurance. I know that's something of a sticky, sore subject with some people here, but until the laws change, it's one of those things you need to abide by).