is an expression used with verbs in Japanese, to indicate that some particular action is very unlikely. It is a combination of the 助動詞（じょどうし） (auxiliary verb) そう, the 副助詞（ふくじょし） (adverbial particle) も, and the い-Adjective
To recap, one of the primary uses of そう is to highlight 様態（ようたい） ‘appearance’. Therefore, そうもない
literally means ‘to not even appear like (A)’. This structure will be used after verbs in their 連用形（れんようけい） (conjunctive form).
My arm hasn't healed yet, so it is very unlikely that I will exercise for another week.
Since the thunder is very bad today, it does not even appear like I can go fishing.
そうもない is very similar to そうにない. The primary difference being that そうもない is stronger. This is due to も expressing something that is surprising (or for emphasis), while に simply marks a location (or state) within which something does or does not exist. The difference can be understood as follows:
(A) そうもない - To not even look like (A), where ‘even’ emphasizes (A).
(A) そうにない - To not be looking like (A), where ‘be’ marks where (A) is ‘not’ existing.
is an expression that is very commonly used with potential verbs, in order to convey that the ‘possibility’ of something cannot be observed. While this translates as ‘to not appear as if (A) can be done’, realistically it comes across quite strongly, similar to ‘to not stand a chance of doing (A)'.
This amount, it doesn't even seem like I can eat it! (There is way too much to even seem possible to eat)
Today, I might not be coming home until 11, so I might not be able to watch the show I always watch.