'Home' in 'Ben and Jen went home.' Can an adverb be a noun at the same time?

In this sentence:

Ben and Jen went home.

Is home both an adverb and a noun?

Home in this construction is short for at (one's) home (with a locative verb) or to one's home with a verb of motion.

• Bill is (at (his)) home now.
• They went (to their) home.

Home is one of those idiomatic locative nouns that are allowed to drop determiners and occasionally even prepositions, e.g.

• be at (the/one's) university/college/school
• be in (the/one's) hospital/university/college/school

So, what it is is an adverbial prepositional phrase that has been stripped of its prepositions and determiners, leaving only a naked noun to stand for the whole phrase. That's how a noun can be used as an adverb.

The tacit assumption in analyzing any sentence is that, for a given analysis of a sentence, every word belongs to exactly one grammatical word class, and there is one unique set of relationships between the words and the phrasal units.

In the case of an ambiguous sentence, there can be two or more possible analyses of a sentence, and so a word could belong to one word class in one analysis, and another word class in another analysis. If the sentence is not ambiguous, there is exactly one possible correct analysis (Though there will usually be debate between competing analyses).

So regardless of the word and the sentence, the answer to the OP's question is no. As another answer here points out, there will of course be cases where a particular word shows up in two different sentences with different functions in each case.

The same word can be a noun and an adverb but in different contexts. For example, consider home.

• I am home.
• He stays home.
• Ben and Jen went home.

In these sentences, home acts as an adverb, telling us the location of the subject. In other words, it qualifies the verbs be, stay and go respectively.

• Home is where the heart is.

In this sentence, home is a noun.

A noun can be used as an adverb as well in cases that do not involve the locative.

 "He stood at the door **hat in hand**." 

Clearly, hat modifies stood by telling the manner in which he stood. Grammatically this is similar to the adverbial function found in "He ran fast."

The OED lists hat as a verb and a noun. It does list hat in hand under phrases and locutions, but does not analyze the grammatical category of the word, hat, in that case.

Of course, one could imagine many more examples: "He stood at the door chair on shoulder,*face aglow*,pockets empty, etc. There is really no limit to such usage of nouns as adverbs. In the right context, English can handle it very well.