As far as I can tell, there are only a limited number of nouns that work with "be of". I would classify them as idioms. I am listing the most common ones I can think of, where commonness is judged by Google Ngrams. I have grouped them into sets of near synonyms. There are
be of use,
be of help,
be of aid,
be of service,
be of assistance;
these idioms mean that something/someone can be used, can help, can aid, can serve, or can assist. There are also
be of relevance,
be of importance,
which mean that something is relevant to/important for the current subject of discussion.
Also,
be of value,
be of benefit;
these mean that something is valuable, usually with respect to the current subject of discussion. Also
be of interest,
which means that something is interesting.
And for a slightly different "be of" idiom, there is
be of age,
which means that someone has reached adulthood.
If you are learning English, it should be fairly easy to figure out what one of these constructions means. But I would suggest that if you use this construction, you should treat it like an idiom, and memorize the specific phrase(s) you want to use. Many nouns describing attributes, like "security" or "tact", cannot be used with "be of". (Google books gives a few instances for "be of security"; it seems to have been used occasionally in the 19th century, but it sounds very strange today.)
“Very out of the way”
It is a bit tough to find cases of very modifying individual prepositions, but it is easy to find cases of very modifying entire prepositional phrases as a unit, just as it does other adjectives and adverbs.
- I think it’s very out of character for him.
- Things can be very out of place.
- Or very out of date.
- And very out of the way.
- They are very on top of music and very adamant about what they carry.
- I just really was very on the edge. Very on the edge, for a while.
- He’s very off his rocker today.
- She is really very behind the times, isn’t she?
- You’re not very with it today, are you?
Sometimes you can use it to mean “very much” or “very far”.
- Are you for Obama? Yes, I’m a little bit for him, just not very for him.
- Are you against the draft? Yes, I’m very against it.
- Is that over your head? Yes, but not very over.
- Walk down the street past my house, but not very past it.
- If you can be more into something, you can be very into it. Is there anything you’re less into?
- No thank you, that whole scene is very beneath me.
At which point, the whilom preposition starts acting more adverbially, and adverbs can be veried without half so much trouble. But some of the directional or adverbial ones really do seem to be modifiable by degree:
- Is the end near? Yes, I believe we’re very near the end now.
- Will the parade pass near here? Yes, it will pass very near us. It shall pass very near, indeed.
Being very like something
However, back to your main point about very like something.
I’m not sure that I’m willing to call like a preposition (the OED calls it an adjective for these sorts of uses), but here are assorted OED citations of “very like”:
- 1719 De Foe Crusoe i. xᴠɪ, ― It was very like the Tree we call Fustic.
- 1710 Swift Jrnl. to Stella 25 Oct., ― Addison’s sister is a sort of a wit, very like him.
- 1727 A. Hamilton New Acc. E. Ind. II. xxxɪx. 81 ― The Durean is another excellent Fruit, but offensive to some Peoples Noses, for it smells very like human Excrements, but when once tasted the Smell vanishes.
- 1868 Yates Rock Ahead II. 245 ― Wooded uplands suggested good cover-shooting; broad expanse of heath looked very like rabbits.
- 1857 R. Tomes Amer. in Japan vi. 135 ― The jamana··is very like the red-wood of Brazil and Mexico.
- 1865 Pall Mall G. 25 Oct. 10 ― Gladiateur’s colours are blue and red, and Nu’s are cerise (which is very like red) and blue.
- 1836 Sir G. Head Home Tour 144 ― The town of Dewsbury··celebrated for··grinding old garments into new; literally tearing in pieces fusty old rags··by a machine called a ‘devil’, till a substance very like the original is reproduced.
- 1926 F. Z. Snoop Reproduction & Sexual Evolution 83 ― Havelock Ellis quotes other cases, even butterflies (if insects may be here included) who possess excrescences on their penes, which of necessity must cause pain, or something very like pain, during coition.
And regarding your original Shakespearian citation, the OED notes that this has become (or was at one time) a set phrase, to be very like a whale:
Allusive, proverbial, transf., and fig. uses of sense 1. a. Prov. phr. (to throw out) a tub to the whale: see tub sb.9 b. very like a whale (after Shaks. Ham. ɪɪɪ. ii. 398): see quot. 1859.
1591 1st Pt. Troub. Raigne K. John (1611) C 3 b, ― The mariner, Spying the hugie Whale, whose monstrous bulke Doth beare the waues like mountaines fore the wind, That throwes out emptie vessels, so to stay His fury.
1859 Slang Dict. 115 ― Very like a whale, said of anything that is very improbable.
So in summary, I think that whatever you call like, there is no problem with things being very like other things.
Best Answer
If it's a formal letter I'd express it as, "I am writing to enquire whether you are still accepting applications for the position of Paper Clip Straightener as advertised on SEEK on 19th July."
If you're in north America, the date should be month then day, so "July 19th".
Enquire and Inquire are essentially the same word, but "enquire" connotes a general question, whereas "inquire" can connote a more formal inquiry.
Since it's formal writing, contractions (such as "it's" for "it is") should not be used unless you're quoting, and avoid superfluous words and improper constructions, so "as to whether" is just "whether", "off of" is "on" and so on.