You don't have to be a gym junky at all and while the gym will help for some cross training it is not typically the type of fitness which you need to ensure that you are confident out in bigger surf.
Paddle fitness
The more you're in the water the stronger a paddler you'll become. Use your arms! A sponge isn't as buoyant as a surfboard so you'll tire more easily than if you're paddling on a surfboard but that doesn't mean you should not use them. Try paddle through the burn a bit and then alternate between paddling with your arms and your legs. You don't want to tire yourself out by sprint paddling each session as you'll want to have energy when you're on the wave but if you're not a good paddler then you want to build up your paddle strength and endurance.
Your aim should be to get yourself to a point where you're confident that you can paddle (or swim) yourself out of trouble. If you're competing, then being paddle fit will enable you to catch more waves than your competitors and there is nothing more embarrassing than not being able to make it out the back for your heat.
Pushing yourself in larger surf
Everyone has their limits - for some "big" is bigger than others. Maybe you want to ride bigger than your mates and maybe that is what will drive you. What you should do however is focus on you and build up your ability and confidence in bigger surf.
Start by making sure you're fit - the more you're out in the water paddling, the more you'll improve your paddle fitness. Include some
cross training (
read this article by Alistair Taylor). When bodyboarding and especially in larger surf, you need bursts of energy e.g. to sprint paddle into a wave or out of trouble. And what you want is to not end up so exhausted that when you are on the wave you can't do anything (again if you're competing this will become so much more apparent). If you are running or swimming to keep fit then to simulate this you need to introduce a few speed bursts e.g. jog 5 lamp posts and pick up the pace for 1 lamp post.
A little exercise which I used to do on the beach, early morning or after a surf, was mark off a start point, a 10m point, 30m and 40m then I would jog the first 10m, sprint 20m, then jog the final 10m, drop down and do 10 push-ups and then turn around and repeat and do this 10 times. This was quite a killer for me but when you get your fitness up you really improve your confidence in bigger surf because you don't get as tired.
All this will help improve your lung capacity, but what you should realise is that you might be able to hold your breath for 2 minutes in the bath but when you've been paddling for what seems ages and couple with that the fact that big surf will trigger your adrenaline which increases your heart rate and it's need for oxygen then you're going to battle holding your breath for a few seconds.
Working specifically on lung capacity, you can try swimming under water or holding your breath for a few seconds while running. I never used to do any of this but there are ways to improve this - Ishmael Grant apparently runs up Table Mountain with a face mask which restricts his air intake.
The important thing to remember when riding bigger surf is to stay calm - RELAX! If you're getting a huge set wave on the head then just accept the beating and trust that you will be coming up within 10 seconds and probably less - count while you're under water 1-mississippi-2-mississippi-3-mississippi-4-mississippi-5-mississippi ... by the time you reach 6 you'll probably have surfaced already and you'll surprise yourself at how short the hold under's are when you do start counting. Just don't panic and you'll breeze through them - if you panic then you're going to send your heart rate through the roof, meaning it will want more oxygen. Try enjoy the beat downs - don't fight it and wait till you're no longer getting thrown around before swimming to the surface.
All in all - if you want to be able to pull into bigger waves then you need to be confident in your ability i.e. you can hold your rail, make the drop and bottom turn in the bigger stuff and when it does come to getting a beating (because you will at some point) that you will be able to handle it.
and if so what do you guys do too keep fit like jog etc