The batteries in old Macs often fail after a few years. Generally, the batteries are good for 1000 cycles, meaning the battery can be completely drained and recharged a thousand times, or half discharged then recharged 2000 times, etc.
After this time the battery might still work, but will supply power for less time. Eventually the battery fails completely and only supplies power for a few seconds or no time at all. And the ultimate failure is probably "bulging battery syndrome" where the battery bulges, sometimes pushing the case open, or jamming against the trackpad.
Bulging batteries should not be used, because there is a small but non-zero chance of a serious malfunction leading to excessive heat, and maybe even a small fire!
Luckily, old laptops can run from a power supply without a battery being fitted. Of course, if you bump the power cable the computer dies immediately and you will lose your work, so you need to be careful.
Some old laptops won't boot this way if the power supply is lower power. These seem to be OK to charge the battery but not enough to start the computer without having the battery power as well. So if you are going to try using this to run your old computer, make sure you have the higher power models.
For example, I recently tried this with a 15 year old 15 inch MacBook Pro. The 60 Watt power supply the owner had been using wouldn't start the machine, but a 76 Watt model I had in my spare parts cupboard worked fine.
It is possible to get third party replacement batteries for these old machines, and they aren't too expensive, but I needed to get it working before the new battery arrived (it was ordered from China).
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