Firstly work out what you want to do !! If all you are doing is running a fridge, some LED lights, charging your detector and phone etc, then Option 1 below. If you want to run a caravan with a microwave or Air/con, then option 2 below. Bear in mind that you will not run the fridge off the generator 24/7.
OPTION 1
Generator - a 1kVA is ample, I would only ever use a Honda or a Yamaha pure sine wave.
Battery - 100AH AGM deep cycle (AGMs are totally sealed and can recover from almost total discharge and need no maintenance.) An AGM battery's life depends on how deep you cycle it. Cycling is defined as drawing current ie 20AH and then recharging, if it is on constant charge it is not cycling. Cycling 10 to 20% of capacity gives approx. 1,000 cycles. Cycling to 50% reduces cycle life to approx. 300 cycles. So don't draw too much too often, or recharge at or about 20% discharge.
A solar panel ie 80 to 120W will extend life, but bear in mind that it only charges with sun and needs to replace what is used overnight.
OPTION 2
Generator - you need a 2kVA again Honda or Yamaha
Battery - Same as option 1 but you could use 2 in parallel giving you 200AH
Solar as per option 1
GENERAL
To charge an AGM you need a smart charger at least 3 stage preferably a 5 stage of 20amps. AGMs don't take kindly to charging to quickly or too high voltage.
Fridge - Personally nothing but Engel, but Evacool or Waeco are other options. Gas fridges are good re gas as said above, but I have one in my van and regulating the temp is a pain on a daily basis.
Myth - "You can charge a battery from a generator." - "You can't". Generators have a 12VDC output, that is all it is, a 12volt source. 12 VDC batteries are charged nearer 14 VDC, so you need to plug a smart charger into the 240 VAC outlet.