This question is not really about what is correct or incorrect. It's about what is most appropriate or idiomatic. And while I would opt for (b) as the most likely answer, it's not the only correct answer. Much depends on the context.
(a) >The bell was ringing and ringing but no one picked up the phone.
This past continuous tense, which emphasises the continuous ringing of the bell by repetition, might well be used by someone telling a story. The speaker tries to keep the action alive by using a tense close to the historic present. It implies that the event is not yet over.
(b) >The bell rang and rang but no one picked up the phone.
The use of the past tense implies that the event is over. This a perfectly acceptable and correct sentence if not quite as neat as (b).
(c)>The bell was ringing repeatedly but no one picked up the phone.
Like (a) this sentence keeps the action alive if a speaker is giving a dramatic account of an event. It is also correct but most likely to be used in a specific context.
For example, it's the tense that a speaker might use to describe a fire, as in:
The fire was burning fiercely but there was no sign of the fire brigade.