What is the grammar behind 没四百下不来
Actually your interpretation is not exactly correct(Although pretty close). The sentence means 400 yuan will be approximately sufficient(still probably a little bit too high or too low) to buy the stuff.

This is quite an oral expression(more commonly seen in northern China). But you can interpret it this way: “拿下来” means "get/fetch/catch something", a connotation which can also be expanded to "buy something"(e.g. get the ownership of something). In this sentence, “下来” can be seen as a shorthand for “拿下来”. And therefore “下不来” is "unable to get", the "不" in the sentence signifying negation(cannot, unable to). And the whole sentence therefore is "Without (at least) 400 yuan you cannot buy it". This should now make sense to you.