Russian has two distinctive features.
The first one is the proximal possession. У меня есть выбор ("I have a choice") literally translates as "there is a choice by (or next to) me". Russian does have a verb which means "to have" but its usage is limited compared to most other Indo-European languages.
The second one is the negation genitive (родительный отрицания). It works slightly differently for subjects and objects, but either way, the accusative or the nominative are in some cases replaced with the genitive in negative sentences:
Произошли изменения (nom.) "changes happened" / не произошло изменений (gen.) "no changes happened".
Я пил воду (acc.) "I drank water" / я не пил воды (gen.) "I didn't drink any water"
Note that in the first example, the voice of the sentence changes as well. The sentence becomes impersonal and the verb changes into neuter singular.
In your example, you can see both features in action.
It is thought that Russian acquired both these features from Uralic languages.