brake seems slack but pad is already close to rim
In addition to all the points Jeff is talking about in his answer, there may be a simple mismatch between your brake lever and the brake: V-brakes require the brake lever to actually pull the cable a fair distance, much more than other brake systems require. As such, if you combine a V-brake with a lever intended for some short pull brake, you will simply not be able to move the brake pads far enough.
The classic cause for such a mismatch is when your bike has been reconfigured from cantilever brakes to V-brakes some time ago, but it may also be caused by someone installing a new lever without being aware of the pull length issue.
If this is the case, the only remedy is to install a V-brake lever. You may check your brake lever whether it is actually configurable to act as a long-pull lever, but most brake levers are not configurable. Thanks to Artefacto for pointing out that there are also pull length converters. These replace the noodle of a V-brake and adjust the amount of cable action.