I think the answer is
I contain what's in a bleaker war,
As pointed out by M Oehm in the comments, "bleaker war" is an anagram of "lawbreaker" so this may be referring to a prison cell which would contain such an individual.
Alternatively, a sleeper cell is a group of agents that remain inactive for a long period and a cell is a group forming a nucleus of political activity, such as a terrorist cell.
Figures entered and values stored.
A cell is a unit in a database table and is a basic unit in flash memory.
Within your eyes I keep the watch running,
Your eye contains cone cells which are responsible for colour vision - they keep the "watch" running (a play on words).
With light I can rise if I let the sun in.
A photocell is a device which uses the photoelectric effect to generate current. Alternatively, this could be referring to photosynthetic cells in plants.
Stored within all that's alive,
A cell is the functional basic unit of life.
When lost and replaced you might just survive.
Cell regeneration is the process of renewal, restoration, and growth that makes genomes, cells, organisms, and ecosystems resilient to natural fluctuations or events that cause disturbance or damage.
Distances covered without moving a fraction,
I think this refers to cell phones (mobile phones), instant communication with far away places.
It's time to break off and take our own action.
I think this refers to a Clandestine cell structure.