There neither exists a spot to stand after the ordeal.
I can't think of a sentence or situation where this works as written. The usual structure is "neither [A] nor [B]" and, while it's permissible to have the nor without the neither, neither must be paired in some way with a nor or a not. While there is a "not" in the first sentence, it's not really paired with the neither in the second, since they're talking about two different things.
Here's how you could write it:
There isn't an appropriate place to break up. There doesn't exist a spot to stand after the ordeal.
This sentence doesn't make any sense to me, because I don't understand how "breaking up" has to do with where you will stand afterwards, but it is grammatical.
Here's an example of one that does work (although it's very informal English):
There isn't an appropriate place to sit around here. Neither can you stand.
In this case "sit" and "stand" are related actions, so the comparison makes sense. You can also say this, which is more "proper":
There isn't an appropriate place to sit around here, nor can you stand.
Edit: As a side note, if I understand your intention correctly, you can write the sentence as:
There's never a "good" place to break up with someone -- but if you can, pick a place where you can make a easy getaway.