The true eels are ostensibly marine fish although some have a freshwater component to their life cycle while fewer still are found in freshwater.
Although most get too large for the average aquarium it is of course possible to house them in an appropriate man made environment if it is suitable.
The lore, legend and science to these utterly fascinating creatures fills many volumes an the reader is encouraged to look up all they can on the history, status and diversity of the true eels. Pay particular attention the one in New Zealand, it's a dinosaur relic as far as eels go. The Anguilliformes were revised by Santini et. al. in 2013 and is reflected herein. The abstract from this paper is reproduced below.
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From Molecular Phylogenetics and Evolution: Volume 69, Issue 3, December 2013, Pages 884-894 - A multi-locus molecular timescale for the origin and diversification of eels (Order: Anguilliformes) Abstract
"Anguilliformes are an ecologically diverse group of predominantly marine fishes whose members are easily recognized by their extremely elongate bodies, and universal lack of pelvic fins. Recent studies based on mitochondrial loci, including full mitogenomes, have called into question the monophyly of both the Anguilliformes, which appear to be paraphyletic without the inclusion of the Saccopharyngiformes (gulper eels and allies), as well as other more commonly known eel families (e.g., Congridae, Serrivomeridae). However, no study to date has investigated anguilliform interrelationships using nuclear loci. Here we present a new phylogenetic hypothesis for the Anguilliformes based on five markers (the nuclear loci Early Growth Hormone 3, Myosin Heavy Polypeptide 6 and Recombinase Activating Gene 1, as well as the mitochondrial genes Cytochrome b and Cytochrome Oxidase I). Our sampling spans 148 species and includes 19 of the 20 extant families of anguilliforms and saccopharyngiforms. Maximum likelihood analysis reveals that saccopharyngiform eels are deeply nested within the anguilliforms, and supports the non-monophyly of Congridae and Nettastomatidae, as well as that of Derichthyidae and Chlopsidae. Our analyses suggest that Protanguilla may be the sister group of the Synaphobranchidae, though the recent hypothesis that this species is the sister group to all other anguilliforms cannot be rejected. The molecular phylogeny, time-calibrated using a Bayesian relaxed clock approach and seven fossil calibration points, reveals a Late Cretaceous origin of this expanded anguilliform clade (stem age ∼116 Ma, crown age ∼99 Ma). Most major (family level) lineages originated between the end of the Cretaceous and Early Eocene, suggesting that anguilliform radiation may have been facilitated by the recovery of marine ecosystems following the KP extinction."
AnguilliformesSuborders in Order