Why 'an old computer' in the following sentence is using 'ancien ordinateur' ?

Xabi, 

the question is asking for the translation of the English word ‘ old ‘ as in meaning ‘ former/previously mine ( used to own ) ‘, rather than as ‘ aged ‘. 

Without the hint or some context specifying clearly which sense of ‘ old ‘ is intended, the English sentence is ambiguous. It is however standard usage at least in my part of the English-speaking world - in everyday speech the meaning would be understood from context.

The point being made precisely by the French sentence, with ‘ ancien ‘ preceding the noun, is the previous/ former ownership of the computer.

To describe both its former ownership and age in French you could add context   ‘ .. j’ai donné mon ancien ordinateur qui n’était pas vieux … ‘

Its because the position of adjectives follow BAGS

B-beauty

A-age

G-goodness

S-size

All BAGS adjectives for e.g belle/beau, grand/petit go before the noun

here, ancien refers to the age (new or old) of the laptop, and so it goes before

I’m interested to know how old something has to be to be ancien? (after the noun). Have personal computers even been in general use long enough to qualify?

Anne, 

I don’t think the answer to your question is specific to French.

 If you would call it ancient, or old-fashioned etc in English, then ‘ ancien ‘ will mostly work fine in French too. On most occasions when we use just plain ‘ old ‘ for something ‘ aged ‘in English ‘ vieux’ will fit well in French. 

See Larousse bilingual entries for more context and examples :

 https://www.larousse.fr/dictionnaires/francais-anglais/ancien/3327

 https://www.larousse.fr/dictionnaires/francais-anglais/vieux/80971

 https://www.larousse.fr/dictionnaires/anglais-francais/old/598779

 https://www.larousse.fr/dictionnaires/anglais-francais/ancient/562340