Well, I am not a definitive expert in modern Russian (in an academic sense), so I will give you the perspective from my highly subjective point of view.
Not that the phrases were outright obsolete, but Chechov’s texts do have distinctively old-fashioned (and even somewhat strange) feel. The way words are used is also quite strange when compared to modern language. The language has drifted substantially towards shorter, more directly phrased, less complex phrases; some words that were used freely now need some helper words to make the meaning clearer, etc. Well, it is still quite understandable and correct Russian. The matter is, no one speaks like this nowadays.
I’ll try to find some examples that are not too long (which is a hard task, given that Checkhov tends to write in longish clauses).
“Если она здесь без мужа и без знакомых”, - соображал Гуров, - то было бы не лишнее познакомиться с ней".
Here, the clause “было бы не лишнее познакомиться” feels quite old-fashioned. Modern language is more determined: “было бы неплохо/было бы здорово познакомиться”. “Было бы не лишнее” is now almost invariably said as “было бы не лишним”.
“Изменять ей он начал уже давно, изменял часто и, вероятно, поэтому о женщинах отзывался почти всегда дурно(…)” - here, “дурно отзываться о ком-то” is somewhat archaic. “Дурно” is a word that is almost never used in everyday speech, and the only phrase where this word can be met today is perhaps “дурно пахнет” (in the sense that something has a foul, unpleasant smell).
“В рассказах о нечистоте местных нравов много неправды, он презирал их и знал, что такие рассказы в большинстве сочиняются людьми, которые сами бы охотно грешили, если б умели(…)” - here we have at least two places where words are used in strange way.
First, the clause “нечистота нравов” sounds peculiar. There is nothing particularly strange about it, it is just that no one speaks like this nowadays. You can find stronger, more direct references to lower morals in modern language, e.g. “разнузданность нравов”. But “нечистота”? That’s weird. The meaning of word “нечистоты” (meaning wastes & faeces) surely affects the feel of the word, because nowadays there is so much talk about environmental problems in the media.
Secondly, in modern Russian the verb “презирать” is used mostly with reference to human beings (of course, I do not mean poetic language here). Chekhov uses this word in reference to inanimate noun “рассказы”, which is quite strange. Modern speakers would phrase it like “он считал их [эти рассказы] чушью”, “не обращал на них внимания” or “он не придавал им значения”.
Sorry, I have to close my speech now. If that is not enough to illustrate my point, I will gladly provide some more examples tomorrow.
If you get through the chapter «Бесконечный спуск» you’ll probably be acclimated to his Russian, and things will become easier. I thought of it as “the endless chapter.”