The sentence Many of them were able to begin buying their own homes contains two clauses, and therefore at least two verb phrases, so asking what "the verb phrase" or "the object" is is problematic.
The first clause in the sentence is the whole sentence. The first verb phrase is:
- were able to begin buying their own homes.
The pair of words were able by itself is not a syntactic unit, and therefore is not a phrase on its own.
The word able in this clause is an adjective that takes an object, the infinitival clause to begin buying their own homes.
The second clause in the sentence is the infinitival clause:
- to begin buying their own homes
Technically this clause has two verb phrases:
- begin buying their own homes
This latter verb phrase contains the verb buying which takes an object, their own homes.
As in the case of were able the strings of words to begin buying or begin buying are not syntactic units, and therefore not phrases on their own.
The word to is not typically considered part of the main verb phrase, since it functions more like an auxiliary verb in terms of its syntactic position.