The correct option is (ii) and (iv).

To understand why sentences (ii) and (iv) are grammatically correct, let us analyze the placement of the adverbs of degree/manner "hardly" and "barely" in relation to the modal verb "could" and the main verb "walk".

In English grammar, adverbs of frequency, probability, and negative limiters (such as hardly, barely, scarcely, and almost) are typically placed in the mid-position of a sentence.
When a verb phrase consists of an auxiliary/modal verb (like could, would, will, has, etc.) and a main verb (like walk), the adverb must be placed after the first auxiliary/modal verb and before the main verb.
The standard word order structure is:
Subject + Modal/Auxiliary Verb + Adverb + Main Verb

Let us evaluate each sentence using this grammatical rule:

Sentence (i): "Raja hardly could walk."
Here, the adverb "hardly" is placed before the modal verb "could". This violates the standard word order, making the sentence grammatically incorrect/awkward.

Sentence (ii): "Raja could barely walk."
In this sentence, the adverb "barely" is correctly placed between the modal verb "could" and the main verb "walk". Therefore, this sentence is grammatically CORRECT.

Sentence (iii): "Raja barely could walk."
Similar to sentence (i), the adverb "barely" is placed before the modal verb "could". This is grammatically incorrect.

Sentence (iv): "Raja could hardly walk."
In this sentence, the adverb "hardly" is correctly placed between the modal verb "could" and the main verb "walk". Therefore, this sentence is grammatically CORRECT.

Consequently, sentences (ii) and (iv) follow the correct grammatical rules of adverb placement, making (ii) and (iv) the correct choice.