Updated: Aug 13, 2019
To provide for the inhabitants, Forever Fleet ships need a stable power source, for propulsion as well as to help societies in climate crises with water, food, energy and production the ships need a great stable power surplus.
Nuclear power has the greatest EROI (Energy Returned On Investment) besides hydro power, meaning that it costs the least resources per unit of energy produced, is scientifically safer than all other energy sources, has as close to zero emission as anything today (and negative emissions when combined with CCS - carbon capture and storage) and has a long and superb track record for use in sea going vessels.
By extracting uranium from the surrounding seawater, the fuel supply is abundant. Nuclear is far from perfect, but as energy source it is currently the scientifically best option we have for Forever Fleet.
Energy.gov has an easily-understandable article about the micro nuclear reactors, probably to be modified for use in Forever Fleet as standard 1MW components - here is the video:
There is however a societal stigma and psychological fear of nuclear power, which may lead to some FF vessels being powered by different means based on their inhabitants' preferences, from solar cells, wind and water to electro-fuels produced by other vessels.