Athena has been extensively rebuilt over the years, so although she has been built in 1948, I wouldn't necessarily consider her "too old".
As for ships in general, it's hard to say what would be "too old":
- when ships no longer meet the safety requirements (double hull for tankers, SOLAS 2010 for passenger ships...), they can be said to be old;
- tankers, bulkers, container ships and such might have a strictly defined fatigue life, meaning that once they become around 20-25 years old, it might be better to scrap them for safety reasons;
- old ships are generally more expensive to run, so the shipowner might consider them "too old" in light of operating expenses;
- the charterer might have something to say about the age of the ship as well, especially in offshore business and oil transportation;
- a passenger ship might be "outdated" from the passenger's point of view - for example most people prefer to have a bathroom attached to their cabin, and cruise ships with no balconies probably have some problems attracting customers in these days...
And so on.