JLPT N1: をおいて (no other but; no one besides)

The construction "NOUN ＋ をおいて" is used when you want to say that there is "no other [something] besides NOUN."

をおいて will often be followed by a word or phrase containing ～ない (i.e. a negative clause).

In particular, the phrase "NOUN をおいて（ほかに）ない／いない" is quite common:

女王役にふさわしい人は彼女をおいてほかにいない。
じょおう　やく　に　ふさわしい　ひと　は　かのじょ　をおいて　ほか　に　いない。
There is no one that is fit to play the queen but her.
Literally: “queen + role + に + suitable / fitting / worthy + person + は + she + をおいて + aside from / other than + there is not.”

I don't know if this is the case, but I like to think of this おいて as being the verb 置く (おく // to put; to place; to leave; to set down; to put aside).

I mean, it kind of makes sense to say "putting aside [something], there is no..."

Right?

If you don't think so, then your only hope is to learn through lots of examples:

ウクライナの歴史を学びたいなら、読むべき本はこの本をおいてほかにないと思います。
ウクライナ　の　れきし　を　まなびたい　なら、　よむ　べき　ほん　は　この　ほん　をおいて　ほか　に　ない　と　おもいます。
In my opinion, if you want to learn about Ukrainian history, then all you need to read is this book.
Literally: “Ukraine + の + history + を + want to learn + if (it is the case that), + should read (=read + べき) + book + は + this + book + をおいて + aside from / other than + there is not + と + think.”

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Maybe you've noticed this by now, but をおいて is used when you are praising a NOUN as being the only [something] suitable for whatever you're talking about.

That is, you're saying "nothing compares to NOUN."

Another example:

これほど子供たちが喜ぶ日は、クリスマスをおいてほかにあるだろうか。
これほど　こども　たち　が　よろこぶ　ひ　は、　クリスマス　をおいて　ほか　に　ある　だろう　か。
Other than Christmas, there is no day when children [my children] are so happy.
Literally: “this much + children + が + are delighted / are glad + day + は, + Christmas + をおいて + aside from / other than + there is + だろう + か.”
Note: We can assume that the speaker is thinking this to herself.

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Last one:

私たちをおいて、このことを知っている人は誰もいないはずだ。
わたしたち　をおいて、　この　こと　を　しっている　ひと　は　だれも　いない　はずだ。
Other than us, there’s no one that should know about this.
Literally: “we + をおいて, + this + thing + を + are knowing + person + は + nobody + there is not + should be (the case that) + だ.”

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Do people use をおいて in everyday Japanese?

Not really in "everyday" Japanese, but you do come across をおいて in books, news programs, and so on. So let's learn it!