Many religions have a history of disallowing women (or men, in some cases) from being religious leaders.
- What level of leadership are women allowed to hold in Hinduism?
- Is there anything significant about the history of women in Hinduism?
Yes woman are allowed to become monks/sanyasinis/rishikas.
Hinduism is probably the only living religion which considers God to be feminine. And in ancient times many vedic rishis were women.
The notable female rishikas who contributed to the composition of the Vedic scriptures are: The Rig Veda mentions Romasha, Lopamudra, Apala, Kadru, Visvavara, Ghosha, Juhu, Vagambhrini, Paulomi, Yami, Indrani, Savitri, and Devayani. The Sama Veda adds Nodha, Akrishtabhasha, Sikatanivavari and Gaupayana.1
Another famous female monk was Andal who is revered by Tamil Vaishnavites.
In today's time, as mentioned by @Pradip Gangopadhyay, Ramakrishna Sarada math and Mata Amritanandamayi are examples.
Yes, women can be leaders in Hinduism. Hinduism perceives women as "Shakti" or strength. As the Sanskrit hymn goes
या देवी सर्वबूतेशू शक्तिरुपेना सन्स्थिता
ya devi sarvabhuteshu, shaktirupena sansthita
This hymn, though dedicated to Goddess Durga, is applicable to all women, or so is believed by Hindus.
Hinduism believes that none can be stronger than the Creator: the Mother being the ultimate creator. Hinduism has instances of female leaders - Gargi, Maitrayee, Arundhati in the Vedic ages. Later ages show instances of Sarada Mata, wife of Lord Ramakrishna. She was worshipped by Lord Ramakrishna himself as "Devi Shoroshi"
Yes, women can be monks in Hinduism. The Ramakrishna Sarada math is run entirely by women monks. Mata Amritanandamayi is a famous monk known the world over as the hugging monk. She heads her own organization.
There are examples of women Sannyasis who have attained Moksha in Mahabharata. I will give here 2 such examples:
Pingala said, ‘…What women is there that regards that Supreme Soul as her dear Lord, even when he comes near? I am now awake. I have been roused from the sleep of ignorance. I am no longer influenced by desire. Human lovers, who are really the embodied forms of hell, shall no longer deceive me by approaching me lustfully. Evil produces good through the destiny or the acts of a former life. Roused (from the sleep of ignorance), I have cast off all desires for worldly objects. I have acquired a complete mastery over my senses. One freed from desire and hope sleeps in felicity. Freedom from every hope and desire is felicity. Having driven off desire and hope, Pingala sleeps in felicity. [Ref: Mahabharata Santi Parva Section CLXXV]
(2) In the same Satya Yuga, a woman of the name Sulabha, belonging to the mendicant order, practised the duties of Yoga and wondered over the whole Earth. In course of her wanderings over the Earth, Sulabha heard from many Dandis of different places that the ruler of Mithila was devoted to the religion of emancipation. ....Sulabha became desirous of a personal interview with Janaka. [ The story is very long. So I will shorten it by saying that Sulabha went to Janaka and entered his mind through her Yoga Power to check if Raja Janaka was truly emancipated. Janaka protested saying that as a mendicant and a woman she should not have entered his body to check whether he has really attained Emancipation. Sulabha gave Raja Janaka a good dressing down. I will quote only a few lines of her retort.]
Sulabha said, 'The contact of one that is emancipated with another that has been so, or Purusha with Prakriti, can not lead to intermingling of the kind thou dreadest. Only those that regard the soul to be identical with the body, and that think the several orders and modes of life to be really different from one another, are open to the error of supposing an intermingling to be possible. My body is different from thine. But my soul is not different from thine. When I am able to realise this, I have not the slightest doubt that my understanding is really not staying in thine though I have entered into thee by Yoga.' [Ref: Mahabharata Santi Parva Section CCCXXI]
If you want to go by scriptures, then, as per Brahma-vaivarta Purāṇa,
aśvamedhaṁ gavālambhaṁ sannyāsaṁ pala-paitṛkam devareṇa sutotpattiṁ kalau pañca vivarjayet
“ ‘In this Age of Kali, five acts are forbidden: the offering of a horse in sacrifice, the offering of a cow in sacrifice, the acceptance of the order of sannyāsa, the offering of oblations of flesh to the forefathers, and a man’s begetting children in his brother’s wife.’
So, sannyasa is forbidden for both men and women in this age of Kali, but under order of a bonafide guru in disciplic succession, one can, because a bonafide guru, who represents God to his disciple, can, in extreme circumstances, for the purposes of preaching, order the disciple to accept sannyasa.
Sannyasa, should never be accepted out of pride, or for peace of mind.