Can block ciphers encrypt messages larger than the key length? I am trying to learn some basic cryptography and I am reading some conflicting statements about block cipher.

My current understanding is that we start with an XOR to obfuscate our data. We can then generate a one-time pad to encrypt our message. The only limitation is that the message length must be less than the length of our single-use pad (making one-time padding impractical for large files).

So we introduce block ciphers which encrypt blocks of plaintext at a time. We generate a key which is essentially a mapping of input to output. An example of a block cipher is AES.

I read that AES would blow up if we tried to encrypt a jpeg file - or any message that is larger than our key size (thus introducing stream cipher). I've also read that block cipher is best used on any known-length message (i.e. any file). The latter makes more sense since we're breaking down messages of any length down into blocks anyway.

Which is true? Am I misunderstanding something?
A block cipher like AES itself cannot be used to encrypt large files. One needs to use some mode of operation to chain a sequence of blocks. The length of this whole "chain", which essentially is the product of the block size and number of blocks, is the length that you can encrypt files with. The key size of the block cipher basically tells you what security level you can get from a "chain" of block ciphers (or a single block cipher of course).