As I read Struck and White on Use of a colon after an independent clause to introduce a list of particulars ... (Chapter 1, Rule 7, McMillan, 1979), it is wrong to use a colon here:
It [a colon] should not separate a verb from its complement or a
preposition from its object. The examples in the left-hand column
[LHC], below, are wrong; they should be rewritten as in the right-hand
column[RHC].
LHC: Your dedicated whittler requires: a knife, a piece of wood, and a
back porch.
RHC: Your dedicated whittler requires three props: a knife, a piece of
wood, and a back porch.
LHC: Understanding is that penetrating quality of knowledge that grows
from: theory, practice, conviction, assertion, error, and humiliation.
RHC: Understanding is that penetrating quality of knowledge that grows
from theory, practice, conviction, assertion, error, and humiliation.
The logic of the first paragraph applies equally well when the particulars are questions.
Either of the following would work:
Option 1: She pushed the thought aside because she didn't want to ask herself questions like [or such as] the following: Where did he go? Was he even real? Was she going crazy? She just wanted to pretend like it had never happened. [Not ideal for fiction. :-)]
Option 2: She pushed the thought aside because she didn't want to ask herself questions like "Where did he go?" "Was he even real?" "Was she going crazy?" She just wanted to pretend like it had never happened. [No colon, quotes to separate the questions from the rest of the sentence. Could have italicized the questions instead.]
Option 2 with quotes or italics may be your best bet.