We can have models to which things from beyond the observable universe are relevant. A simple, 'every-day' example: we can model a system to see what something in the system will look like in the future. However, that doesn't mean that the future-thing can be said to 'exist' for us (at least not yet!).
What does it mean to "exist"?
I would propose this:
If it is physically possible for information about something to reach an observer within a certain time frame, then we say that this thing physically exists for that observer in the context of that time frame.
Or (rephrased, but intended to express the same idea):
If for all events that can possibly occur in a given time frame, at least one event results in information travelling from a thing to an observer, then that thing exists to that observer in the given time frame.
Examples:
'Walkmans (..Walkmen..?) existed in my life time.'
What it really means: More than one Walkman were so positioned in the universe that I could have observed them within my life time.
'There were no computers back in the 16th century.'
What it really means: Throughout the period of time we call the 16th century, no one could have observed a computer.
What we omit (and assume) in the sentence: The identity of the observers (presumed to be 'all humans')
'Dragons never existed'
What it really means: From the beginning of time to the present (we will ignore the difficulty in defining 'present' for now), no one could have observed dragons.
What we omit (and assume) in the sentence: The identity of the observers (all humans)
'Mammoths don't exist anymore'
Meaning: At present, no one could observe mammoths.
Omitted: Observers, time frame (presumed to be some period of time that can be reasonably called 'present'; exact inteval depends on subject of discussion)
'Nanobot soldiers don't exist.'
Meaning: At present, no one could observe nanobot soldiers. (but may be possible in the future)
Omitted: Observers, time frame