Punctuating a phrase leading up to a question

Solution 1:

I believe that 1) is the best choice. According to this source, a colon is used "before a list or an explanation that is preceded by a clause that can stand by itself."

While 5) also seems correct, and the others are at least defensible, I think 1) works best.

Solution 2:

Regarding each of your offered choices:

1. Let’s ask ourselves: what would a world without computers look like?
   This one isn’t too terrible. It might be one of those places where you may wish to follow the colon with a capitalized word, since you are introducing a complete independent sentence. It depends on your publisher’s internal style guide whether to do that.
2. Let’s ask ourselves – what would a world without computers look like?
   This isn’t really my favorite use of a dash.
3. Let’s ask ourselves . . . what would a world without computers look like?
   This is certainly the worst possible choice. That isn’t want an ellipsis is for. It looks like some ten-year-old’s text message ... you know ... the people who know no punctuation than an (unspaced) ellipsis ... so that is all they use ... it looks horrible.
4. Let’s ask ourselves, what would a world without computers look like?
   Not my first choice, although not just crud like the previous one.
5. Let’s ask ourselves, “What would a world without computers look like?”
   If you have to choose from one of these five, then this is the best choice.

However, given my druthers, me I would rather write:

• Let’s ask ourselves what a world without computers would look like.

Because it is now an indirect question, there is no need of terminating question mark, either. But that wasn’t one of your choices.