Quick summary of my research on batteries...
It's recommended to only discharge lead-acid batteries to 50% to get full-life. So cut all those Ah ratings in half! That 12 v, 125 Ah battery might only deliver 800 Watt-hours. They're also for a 20 hour rate, and loads are often higher than C/20.
So I went looking at Lithium Ion. In theory they make lead-acid look obsolete. I figure I could find parts to make a 800 Watt-hour battery for half the price of a new, deep-cycle lead-acid. Of course, that's not counting the cost of my time. Still I was thinking it might be worth it for portable applications, where the pack would 1/10th weight or less. There are 3 big problems with this though. 1) Safety concerns. The packs have to be designed very carefully and properly. 2) I haven't found off-the-shelf packs at a good value. Not sure why. So I guess they are DIY only at this point. 3) General issues with the supply quality and authenticity. This has a bearing on making them safe.
Most Li-Ion cells, (the best value ones anyway) are the "18650" size (18 mm diameter, 650 mm long). They have gotten amazingly cheap, for instance on ebay. It's essential though to find a good supplier, with a return policy, and then verify what you receive. If you believe youtube, there is a lot of battery fraud out there.
On ebay I was surprised to not find any cells with tab connections on them (for making packs), but nevertheless there are plenty of battery holders available.
Wiring 4 in series would give about the right voltage (14.4 - 16.8v). This charging voltage might even be better matched to "12 v" solar panels, which are really 18 volts at max power. (Make sure you use proper charging methods.)
Cell types range from 4 to (supposedly) "8.8" Ah/cell. For more charge capacity (Ah), one would wire these strings of 4 in parallel. 10 strings (40 cells total) could deliver comparable energy (in theory) to the "125 Ah" lead-acid battery mentioned.
For safety, you must use "protected cells" to minimize risk of fire or exploding cells. These "protected cells" are "supposed to" have a tiny circuit board packaged inside the bottom each cell to disconnect the battery from the load if the cell voltage goes to low, and to disconnect the battery from the charger if the charging voltage goes to high.
The temperature of the pack also must be kept within range, and the cells should all be the same type, age, and same temperature so that they behave uniformly. For safety, and to verify it works, I would test the voltage on every single cell at least through the first charge cycle, and then again at some maintenance interval. The pack should have a fuse, and thermal-cutout. Don't charge it below freezing. It sounds like a lot of stuff, but this is basically what is inside every single laptop battery pack. (One difference is that in those, the protection circuitry is outside the cells, not inside. If battery quality can't be trusted, then this may be a better strategy, but it could also be too complicated for DIY.)
Yes there are hazards with any of this and it's a big PITA, but lead-acid batteries are no picnic either. I can't recommend this to an amateur, but if you know about electricity, and know how to be safe.... use your best judgement.
There are kids on youtube electrifying their bicycles this way, probably to the horror (or ignorance) of their mother's.
Good luck.