Origins
The word 'attain' entered English around 1300 from Old French 'ataindre' (later 'atteindre'), meaning 'to reach,' 'to achieve,' or 'to touch upon.' The Old French word has a complex Latin pedigree: it blends two distinct but related Latin verbs, 'attinēre' (to hold to, from 'ad-' + 'tenēre') and 'attingere' (to touch upon, from 'ad-' + 'tangere'). The blending occurred in Vulgar Latin, where the two verbs — similar in sound and overlapping in meaning — merged in the popular form *attangere. The result gave English 'attain,' a word that combines the idea of reaching or touching something with the idea of holding onto it.
The PIE root *ten- (to stretch) underlies 'tenēre' (to hold, originally 'to stretch out to grasp'), connecting 'attain' to the vast '-tain' family: 'sustain,' 'obtain,' 'maintain,' 'pertain,' 'contain,' 'retain,' 'detain,' and 'entertain.' Among these siblings, 'attain' emphasizes the reaching — the stretching toward a goal that is not yet in hand. One attains a summit, attains a qualification, attains a certain age. The word implies effort and distance: the thing attained was not close or easy.
The derivative 'attainment' (the act of achieving, or the thing achieved) has been in English since the fifteenth century. 'Attainable' (able to be achieved) and 'unattainable' (beyond reach) followed naturally. 'Unattainable' carries a particular poetic resonance: the unattainable goal, the unattainable beloved, the unattainable ideal.
French Influence
The legal word 'attainder' — as in 'bill of attainder,' a legislative act declaring a person guilty without trial — comes from the same Old French 'ataindre' but in a different sense: to convict, to reach with a judgment, to touch with a stain. 'Attaint' (to stain, to corrupt, to convict) is a doublet. The idea is that a conviction 'reaches' the person and 'touches' them with guilt. This legal sense was important enough that the United States Constitution specifically prohibits bills of attainder (Article I, Sections 9 and 10).
The semantic range of 'attain' in modern English is narrower than its medieval range. Middle English 'attain' could mean to reach, to achieve, to convict, to affect, and to touch. Modern English retains mainly the first two: to succeed in achieving ('she attained her goal') and to reach a level ('he attained the age of ninety'). The legal and physical senses have faded.
The word occupies an interesting position on the formality spectrum. It is more formal than 'reach' or 'get' but less formal than 'accomplish' or 'achieve' in many contexts. 'Attain enlightenment,' 'attain mastery,' 'attain one's full potential' — the word tends to appear with elevated or abstract objects, reinforcing its sense of striving toward something high.