Can this sentence also mean she is just (recently turned) 70 or just that she is only 70 (not older)?
It would only be taken to mean the latter.
For the former, you could say "Hun er knapt (barely) 70" or "Hun har akkurat fylt 70".
Jeg er bare sytti, meg selv.
bare sytti...det er gammelt nok at hun kunne ha været barnevakt for Master Yoda!
Det er gammelt nok til* at hun kunne * vært* barnevakt for Mester* Yoda
Can we say "hun er kun sytti" ?
Yes, if this weren't a listening exercise that would be equally correct.