It is reasonably safe to leave an AES encrypted RSA private key public? The owner of the RSA key can't store the key, so they need to hand the key to an untrusted entity for storage. Before they hand it over they encrypt it using AES with a multi-word phrase they can remember. 

The idea is that I could generate a private key and store it outside the entity in a non-secure location but not worry (as much) about it since it is protected by a passphrase.

The private key might be stored in several places - in fact and I have to assume they are all publicly-accessible (like a web page). But when the entity wants to do something with the key it requests it back from the storage places and then (with the users password) decrypts it and uses it for what is needed.
Based on skimming the relevant RFCs, I can't find any outright problems. Key derivation from the password is performed via PBKDF2, and best I can tell, the choice of cipher is left up to the implementation. As long as a good cipher is chosen, a truly random 30-character passphrase consisting of uppercase letters, lowercase letters, and digits should have roughly 178 bits of entropy. This is significantly higher than that of a 128-bit key, which is considered to be secure.

It still gives me a bad feeling that I can't shake. But as long as keys are only ever decrypted on trusted devices, I can't personally find a reason why it would be unsafe.

Not that this should be considered an endorsement by any stretch of the imagination.