You are not going to get good light from a 12v battery, at least not for long. Probably best would be a few 12v florescents as used in caravans, for general lighting, with a 12v LED light for more local use in the area you are working on.
Best not to expect the car's own battery to supply the current - use a leisure type battery which is designed to be regularly discharged. You would need to take that in the house to charge it, unless you could arrange a solar or wind charger.
In the dim and distant past and in similar circumstances, I used a couple of paraffin pressure lamps to provide light (and some heat), but not a good idea when messing near petrol.
Following advice here, I would strongly recommend using normal mains fittings with 12 volt CFL bulbs, also available from caravan suppliers. I have these in a stables, run off a standard leisure battery. I use batten fittings inside and various exterior weatherproof lights outside. I use a normal switched spreader with a 5 amp thermal trip from CPC between the battery and the spreader, then 13 A plugs and 0.75 mm "lighting" flex. You might chose to rely on the plug fuses.
The 12 volt CFL gives very similar light levels to the equivalent mains one.
The other type of mobile light which I have found very effective is one of these:
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it provides a directional but wide, diffuse beam. It used mains bulbs but when the inverter dies (which it will) you can wire the socket direct to the battery and use another 12V CFL.
If you can leave the doors open, have a work area near the doors and work during daylight then a battery powered inspection lamp might be sufficient.
But for general lighting IMHO it is a waste of time pissing about with LED's (Point source meaning lots of shadows, and the need to spend a fortune to get decent light levels, plus poor colour rendition and lots of waste heat)
A number of years ago I went through a similar exercise, not for a garage but for offgrid lighting in a *very* remote location with near zero possibility of natural daylight. Linear fluorescents were the right choice then and almost certainly still are. Compact fluorescents are universally hopeless in all circumstances so don't be tempted to use them, you'd get more light be rubbing a couple of glowworms together.
So convert conventional fluorescent lamps (strip lights) to 12v DC operation, run off a battery suitable for deep discharge, charge up at home or bang a solar panel on the roof and fit a charge controller.
Don't even think about buying a caravan / boat 12v fluorescent for general lighting because they will be overpriced flimsy Chinese s**te.
Google "12 dc fluorescent ballast" for a few 'expensive' options, or maybe these from ebay for 11 quid
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(plus about the same again for a 4ft fitting from a local electrical wholesaler)
I've used these ballasts before but they are not readily available in the UK
Depends on what you need. I find a single 8w cfl is plenty for mucking out a 12 ft x 12 ft stable, and that's under a black onduline roof with no reflector. Not ideal for reading fine print. For close up work under a bonnet or inside a car you will need some sort of lead light or torch.
The Lidl and Aldi LED 'wand' lights are reasonably priced, provide a good light, and can be charged from 12V or 240V sources. However they won't provide as much light as a couple of large mains flourecent tubes. See other comments on converting mains tubes to 12V operation.
eBay, chinese suppliers direct. I've recently been building market stall d= isplay lights, based on outdoor rated 5W (9 LEDs in a square array) floodli= ghts that are a _very_ solid turned aluminium case about 2" across and run = directly from 12V. Cost under a tenner each. A couple of gel lead acids (o= ne spare) run them for a whole day.
Cabling is cheap speaker cable - 30-something strand. I'm using 2 pin DIN c= onnectors, as I also have some Ikea desklamps wired into the same system.
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