It sounds like you have several runs of Cat5 coming out of your house into a box. Those runs originate in several rooms of the house, like the living room, bedrooms, etc. So, this is properly wired except for where the wiring closet is supposed to be. If that's not correct, ignore the following...
My recommendation is to (briefly) pretend this was done correctly. Build a proper wiring closet inside the house somewhere, forward whatever your internet provider gives you to that wiring closet, place all the routers/switches/etc INSIDE. Once that's done, you'll need to connect each of the original wires to the switches, so run cable back to the box outside, and install a small patch panel. This will minimize the amount of hardware outdoors, and in that tiny service box.
I would encourage you to re-wire completely as much as possible, so it doesn't loop back outside. But that could be done over time later, if needed. Also, anything that can be wireless reduces the need for cable. For simple, cheap solutions, this could cost less than $50 (cheap patch panel, and several runs of 6' Cat5 just to go through the wall to a wiring closet immediately opposite the exterior box). Odds are you can't get a full panel which would fit in the service box, so look for something you can cannibalize 4-6 port segments out of and mount some other way. For that matter, if it's just a few wires, use a few unsupported keystone jacks and call it a day. ^_^
Finally, I would also recommend you move the cable box about 1.5 feet sideways, and put your own box in this location on the exterior wall. Service providers sometimes get upset if you're poking around in their boxes and you'll definitely butt heads if they want to lock the box to prevent you from tampering, and you want to lock the box to prevent random people from stealing hardwired internet from you. Better to have distinct work areas.