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Lorsqu’un anglophone exprime l’idée far from … home, un francophone l’exprime par loin de chez ….

Seul un jeune enfant peut dire parfois “Je suis loin de ma maison” car il se rattache à l’objet maison qui pour lui est un refuge sécurisé, quelque chose qu’il peut toucher, alors que formuler chez moi demande une capacité d’abstraction plus avancée.

De plus loin de chez … renvoie à l’habitant de la maison et non à la maison elle-même.

Il est plus naturel de dire aussi je suis loin de chez moi quand je pense à mon appartement ou à tout autre habitation qui ne s’appelle pas maison.

Grammatical rules aside, the distinction between “de chez lui” and “de sa maison” is actually very similar to the distinction between “from (his) home” and “from his house” in English.

“From his house” would have specifically referred to the building (and the type of building), hence would have been translated as “de sa maison” (this is very similar to English), whereas “from his home” refers to the place to live, the domicile, hence “de chez lui” is more appropriate.

This parallel is quite generally applicable, but there are of course exceptions, for example:

• “Devoirs à la maison” means “Homework“.
• “Fait-maison” means “Home-made.

Note that you can also use “chez-soi“ (with a hyphen) as a noun, so “loin de son chez-lui” could also have been a possible translation, but certainly less natural here, since “son chez-lui” puts more emphasis on the fact that the home is rather a private place, some sort of retreat (possibly closer to “far from his little cosy home / place of comfort“).

“Loin de chez lui” is correct, and I would suggest to use this one, but a slightly more formal translation could have been “loin de son domicile” (“domicile” isn’t quite as formal in French as in English, but it is a little more formal than “chez lui” nonetheless).

What I think is that, here we need a nominal part on which we can say something is far from it. But chez lui seems to be an adverbial phrase.

This is a very astute observation; the complement of de is usually nominal, and chez lui is indeed adverbial.

However, de chez lui is perfectly correct and idiomatic, and is far from unique; there are many other examples of de taking an adverbial complement: d’ici, de là, d’où, de dehors, de loin, d’en haut, etc., etc., etc.

(The same happens with English from; for example, we never say *”at here” or *”at afar” or *”at inside the box”, but “from here” and “from afar” and “from inside the box” are all perfectly natural.)

“Maison” in French really means a house (with a roof, etc.)

The only way to express a generic home (whether it’s a house, an appartment or whatever) is to use “chez“.

I only want to add if I can, that the meaning of “chez” is originally more complex, deriving from the Latin casa = house, without the subtility of the cases (declension) in modern French: “de chez” is sometimes put for “from the house of” ab casa “je vais chez…” “ad casam” “to the house of…”, etc. “je suis chez moi” in casa, in my own house, i.e. home in English.

The meaning of ‘chez’ covers intrinsically different movements “to”, “in”, “into” or “from”, singular or plural, according to the context of the sentence : chez eux, meaning in their house (of them : casa, casae, casam of several people) of their houses (casae, casas, casis, casarum… several houses of several people) for example in the Jacques Brel’s song, chez ces gens-là…
So we can also say “son chez-soi” his own house (reflective).
Note that casa is an old Latin term meaning a little house, a little farm or a barrack instead of domus, a more formal and greater building, or villa, a greater farm, a little colony or aedes more official.

Case has more a less the same informal meaning in modern French: a little house of an indigenous in Africa, but is more formal in Spanish or Portuguese equivalent of maison in French. Casar in Spanish for exarple is to marry, ‘se caser’ to be married with the same meaning is colloquial in French.

A few examples that may help:

• La boulangerie était loin de sa maison mais près de son bureau. (strictly speaking of the physical building)

• Loin de chez lui, il se sentait seul. (Away from home, he was lonely)

• Il ne se sentait pas chez lui dans sa propre maison. (He was a stranger in his own house)

• Quand il était à la maison, il ne se sentait pas chez lui. (When at home, he did not feel home)

• Après un long voyage de six mois en Asie, il avait hâte de rentrer chez lui. (After six month traveling in Asia, he was eager to go home). Chez lui, in this case, can be his house, his city, his country, wherever he feels “at home”.

In French, maison means “home” in the expression “à la maison”.

“Faites comme chez vous” translates “Make yourself at home”.

“Chez lui” a un contenu affectif. “À la maison” est plus neutre.

I have also heard French people in UK say ‘chez sainsbury’ when they plan to meet there. Also, chez le dentiste etc.

An interesting (and IMO, relevant) example that contains two “chez” constructions, side-by-side, being used clearly as nouns and that illustrates, in my opinion, how important perspective can be, as mentioned in a comment by @ruakh, in determining whether “chez” is conveying the sense of “home” or the sense of “house” is the following proverb:

Un petit chez * soi vaut mieux qu’un grand chez * les *
autres

(also stated as):

Mieux vaut un petit chez * soi qu’un grand chez * les * autres,

(please see the closing parenthetical for explanation of all the *****)

which I translate to mean:

It’s better to live in one’s own small home than in someone else’s
large house.

(or perhaps):

It’s better to have one’s own small home than to stay in someone else’s large house.

*********(I previously thought [and probably still do] that, as mentioned in Bruno’s and Docjipi’s answers, “chez-soi” and even “chez-les-autres” should be hyphenated, but the only hyphenated version of this proverb that I found was on fr.wictionary and N-grams generates slightly more hits without the hyphen for “Un/un petit chez soi” and no hits at all for “chez-les-autres” with hyphens [please re-click the blue button once loaded]. En plus, et bien entendu ça ne vaut pas une référence au GRAND JACQUES comme dans la réponse de Docjipi, même La Fouine laisse tomber les traits d’union!)