Хочется чего-то невозможного is missing from your excerpt indeed.
This phrase means something along the lines of:
Really, don't you too have some kind of angst mixed in with your enjoyment of nature, as if one were yearning for something impossible and missing something bygone?
This phrase, as is the custom for Tolstoy, is clumsy, albeit perfectly understandable in the original Russian. I'm not sure I'm able to convey this clumsiness in English, but here are some things that sound off to me:
Неужели и у тебя, as it is written, means more of are you too one of those people (who aren't feeling the angst) rather than aren't you feeling the angst (like I do). From the context, we can see that the latter reading is the intended one, but, taken literally, it should be the former.
какая-то тоска is rarely used in negative sentences like this one, it should have been никакая ("aren't you feeling any angst") or некая ("aren't you feeling some kind of angst").
жаль чего-то прошедшего literally means "I feel sorry for something bygone", as in "something bygone is hurt and I'm sympathizing with it". It is rarely used with inanimate objects. If the intended meaning is "I'm missing something bygone" or "I regret something bygone", it's usually put as я жалею о чём-то прошедшем or я сожалею о чём-то прошедшем. This one, though, makes the zeugma work, so it's well within the limits of any but the most strict and formal style.
As many have mentioned, this kind of spit-and-duct-tape sentence composition, intended or not, is a perfect device for conveying the stream of consciousness of a person on an emotional roller coaster, like the protagonist of "Family Happiness".