Short answer
No. Raw or rare meat from a halal animal that was properly slaughtered is not haram, in the Hanafi school or the others. The blood that is forbidden is the blood that flows out at slaughter; what stays in the meat is forgiven. Avoid raw meat only when it is likely to make you ill.
Raw or rare meat of a zabiha animal is mubah. Only the flowing blood (dam masfuh) is impure and forbidden; the blood left in the flesh and veins after slaughter is pure. Food that is likely to cause real harm should be avoided.
The four schools agree on this.
Explanation
The question comes up with steak tartare, carpaccio, Lebanese kibbeh nayyeh, raw liver in some cuisines and steak served rare. Cooking is not a condition for meat to be halal. What makes meat halal is the species and the slaughter, and both are settled before the cooking starts. The four schools agree.
Many people worry about the red juice in a rare steak. The Quran forbids blood that is spilled out, and the scholars explain this as the blood that pours from the animal at slaughter. That blood drains away. What remains in the meat cannot be removed and is forgiven; much of the red liquid in a rare steak is in fact water and a muscle protein called myoglobin. Liver and spleen, which are made largely of blood, are named in a hadith as lawful.
The Hanafi books do list seven parts of a halal animal that are not eaten: the flowing blood, the male organ, the testicles, the female organ, the glands, the bladder and the gall bladder. Ordinary muscle meat, cooked or raw, is not among them.
Health is a separate consideration. Islam forbids harming yourself, so do not eat raw meat that carries a real risk, such as raw chicken, raw minced meat from an unknown source, or raw meat during pregnancy against medical advice. And raw meat that was not slaughtered properly stays haram however it is prepared (see non-halal meat).
The evidence
Quran, Surah Al-An'aam 6:145
قُل لَّآ أَجِدُ فِى مَآ أُوحِىَ إِلَىَّ مُحَرَّمًا عَلَىٰ طَاعِمٍۢ يَطْعَمُهُۥٓ إِلَّآ أَن يَكُونَ مَيْتَةً أَوْ دَمًۭا مَّسْفُوحًا أَوْ لَحْمَ خِنزِيرٍۢ فَإِنَّهُۥ رِجْسٌ أَوْ فِسْقًا أُهِلَّ لِغَيْرِ ٱللَّهِ بِهِۦ ۚ فَمَنِ ٱضْطُرَّ غَيْرَ بَاغٍۢ وَلَا عَادٍۢ فَإِنَّ رَبَّكَ غَفُورٌۭ رَّحِيمٌۭ ﴿١٤٥﴾
Say, "I do not find within that which was revealed to me [anything] forbidden to one who would eat it unless it be a dead animal or blood spilled out or the flesh of swine - for indeed, it is impure - or it be [that slaughtered in] disobedience, dedicated to other than Allah. But whoever is forced [by necessity], neither desiring [it] nor transgressing [its limit], then indeed, your Lord is Forgiving and Merciful."
Forbids blood that flows out, not the traces left in meat.
Quran, Surah Al-Baqara 2:195
وَأَنفِقُوا۟ فِى سَبِيلِ ٱللَّهِ وَلَا تُلْقُوا۟ بِأَيْدِيكُمْ إِلَى ٱلتَّهْلُكَةِ ۛ وَأَحْسِنُوٓا۟ ۛ إِنَّ ٱللَّهَ يُحِبُّ ٱلْمُحْسِنِينَ ﴿١٩٥﴾
And spend in the way of Allah and do not throw [yourselves] with your [own] hands into destruction [by refraining]. And do good; indeed, Allah loves the doers of good.
The basis for avoiding food that is likely to harm you.
Sunan Ibn Majah 3314
Graded sahih... أحلت لنا ميتتان ودمان ...
"Two dead things and two bloods have been made lawful for us."
The two bloods are liver and spleen, showing that blood within the body's tissue is lawful.
Where the ruling is recorded
- Radd al-Muhtar (Ibn Abidin), Kitab al-Taharah, Bab al-Anjas. Blood remaining in the meat and veins after slaughter is pure.
- al-Fatawa al-Hindiyya, Kitab al-Dhaba'ih. The seven parts of a slaughtered animal that are not eaten.
Checked by our team. For a more accurate answer to your own situation, consult a scholar. Quran translation by Sahih International. Hadith Arabic from the standard editions of the six books and the Muwatta, with gradings as recorded there; the English is our own rendering. Spotted a mistake? Tell us.