What does 馬鹿 Baka mean?

Baka (馬鹿) means fool or idiot. It can insult someone, but the relationship and delivery can also make it affectionate teasing; it is not reliably mild.

The characters do not supply the definition

馬 is horse and 鹿 is deer. Shogakukan's Digital Daijisen identifies the spelling as ateji, characters used for their sound. A literal horse-and-deer translation therefore misses the ordinary word.

The same dictionary gives a Sanskrit account of the origin. The selected Nihon Kokugo Daijiten entry on that page offers Sanskrit derivations too, with qualified wording. These are dictionary accounts, not a settled history established by the characters themselves. The old story about someone calling a deer a horse cannot, by itself, prove how the Japanese word arose.

An insult with several uses

Daijisen includes foolishness, pointless conduct, an excessive degree and something that no longer functions. It also explicitly notes affectionate uses. Its regional note says baka is more widespread and used especially in Kanto, while aho is used especially in Kansai; that observation does not establish a universal ranking of how offensive each word is.

A dismissive judgment about another person and a complaint about a broken object are different acts, even when the same word appears in both. Calling the root a mild word would conceal that distinction. The target, the surrounding sentence and the speaker's manner all matter.

What the anime scene has to establish

A tsundere comedy can make the insult part of a performance of concealed affection. That is a reading of the relationship in the scene, not an affectionate meaning guaranteed by baka. A hostile exchange can use the same word without the joke.

Subtitles also have to choose an English insult with its own register. Fool, idiot and a stronger alternative do not sound interchangeable. A comparison needs the actual line and translation; this dictionary review does not certify every anime subtitle or dub that uses one of those words.

The useful correction

Baka does not mean horse-deer, and it does not have one fixed level of rudeness. The dictionary supports several senses and a range of interpersonal uses. The scene supplies the rest. Treating a familiar comic delivery as a rule for every occurrence mistakes recognition of a performance for knowledge of the word.