Today food is not a major problem (easy enough to source jellyfish and substitutes can be used), but their extremely pelagic behavior is a huge problem and the reason adults never have been kept successfully for a longer period in an aquarium. Few cases have been attempted. Some weren't healthy to begin with and rapidly demised. Even seemingly healthy animals don't manage, as they end up rubbing their flippers/body on the edges, eventually causing serious infection. Juveniles/immatures are apparently easier to keep and have been kept for a longer period. A few years ago, scientists were able for the first time to keep a larger leatherback alive in captivity by fitting it with a harness that suspended it in the middle of a pool, even though it swam all the time. Not really something that would be suitable for display in a public aquarium!
*
Harnessing Turtle Power - UBC Reports
*
https://www.sciencedaily.com/releases/2008/10/081027204545.htm
*
www.seaturtle.org/pdf/JonesTT_2011_JExpMarBiolEcol.pdf (see "Animal husbandry", page 85)
etc
Similar techniques have been used for shorter periods to keep stranded leatherbacks alive until they can be released.